OPERATIONS RESEARCH
Mathematics and Models
Volume 25
PROCEEDINGS OF
SYMPOSIA IN
APPLIED MATHEMATICS
AMERICAN MATHEMATICAL SOCIETY

PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS
VOLUME 1 NON-LINEAR PROBLEMS IN MECHANICS OF CONTINUA
Edited by E. Reissner {Brown University, August 1947)
VOLUME 2 ELECTROMAGNETIC THEORY
Edited by A. H. Taub {Massachusetts Institute of Technology, July 1948)
VOLUME 3 ELASTICITY
Edited by R. V. Churchill (University of Michigan, June 1949)
VOLUME 4 FLUID DYNAMICS
Edited by M. H. Martin {University of Maryland, June 1951)
VOLUME 5 WAVE MOTION AND VIBRATION THEORY
Edited by A. E. Heins {Carnegie Institute of Technology, June 1952)
VOLUME 6 NUMERICAL ANALYSIS
Edited by J. H. Curtiss {Santa Monica City College, August 195 3)
VOLUME 7 APPLIED PROBABILITY
Edited by L. A. MacColl (Polytechnic Institute of Brooklyn, April 1955)
VOLUME 8 CALCULUS OF VARIATIONS AND ITS APPLICATIONS
Edited by L. M. Graves (University of Chicago, April 1956)
VOLUME 9 ORBIT THEORY
Edited by G. Birkhoff and R. E. hanger (New York University, April 1957)
VOLUME 10 COMBINATORIAL ANALYSIS
Edited by R. Bellman and M. Hall, Jr. (Columbia University, April 1958)
VOLUME 11 NUCLEAR REACTOR THEORY
Edited by G. Birkhoff and E. P. Wigner (New York City, April 1959)
VOLUME 12 STRUCTURE OF LANGUAGE AND ITS MATHEMATICAL ASPECTS
Edited by R. Jakobson (New York City, April 1960)
VOLUME 13 HYDRODYNAMIC INSTABILITY
Edited by R. Bellman, G. Birkhoff, C. C. Lin (New York City, April 1960)
VOLUME 14 MATHEMATICAL PROBLEMS IN THE BIOLOGICAL SCIENCES
Edited by R, Bellman (New York City, April 1961)
VOLUME 15 EXPERIMENTAL ARITHMETIC, HIGH SPEED COMPUTING, AND
MATHEMATICS
Edited by N. C Metropolis, A. H. Taub, J. Todd, C B. Tompkins (Atlantic City and
Chicago, April 1962)
VOLUME 16 STOCHASTIC PROCESSES IN MATHEMATICAL PHYSICS AND
ENGINEERING
Edited by R. Bellman (New York City, April 1963)

VOLUME 17 APPLICATIONS OF NONLINEAR PARTIAL DIFFERENTIAL
EQUATIONS IN MATHEMATICAL PHYSICS
Edited by R. Finn (New York City, April 1964)
VOLUME 18 MAGNETO-FLUID AND PLASMA DYNAMICS
Edited by H. Grad (New York City, April 196S)
VOLUME 19 MATHEMATICAL ASPECTS OF COMPUTER SCIENCE
Edited by J. T. Schwartz (New York City, April 1966)
VOLUME 20 THE INFLUENCE OF COMPUTING ON MATHEMATICAL RESEARCH
AND EDUCATION
Edited by /. P. La Salle (University of Montana, August 1973)
VOLUME 21 MATHEMATICAL ASPECTS OF PRODUCTION AND DISTRIBUTION OF
ENERGY
Edited by P. D. Lax (San Antonio, Texas, January 1976)
VOLUME 22 NUMERICAL ANALYSIS
Edited by G. H. Golub and J. Oliger (Atlanta, Georgia, January 1978)
VOLUME 23 MODERN STATISTICS: METHODS AND APPLICATIONS
Edited by R. V. Hogg (San Antonio, Texas, January 1980)
VOLUME 24 GAME THEORY AND ITS APPLICATIONS
Edited by W. F. Lucas (Biloxi, Mississippi, January 1979)

OPERATIONS RESEARCH
Mathematics and Models

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PROCEEDINGS OF SYMPOSIA
IN APPLIED MATHEMATICS
Volume 25
OPERATIONS RESEARCH
Mathematics and Models
American Mathematical Society
Providence, Rhode Island
1981

LECTURE NOTES PREPARED EOR THE
AMERICAN MATHEMATICAL SOCIETY SHORT COURSE
OPERATIONS RESEARCH:
MATHEMATICS AND MODELS
HELD IN DULUTH, MINNESOTA
AUGUST 19-20, 1979
EDITED BY
SAUL I. GASS
The AMS Short Course Series is sponsored by the Society's
Committee on Employment and Education Policy (CEEP). The
Series is under the direction of the Short Course Advisory
Subcommittee of CEEP.
Library of Congress Cataloging in Publication Data
American Mathematical Society Short Course,
Operations Research: Mathematics and Models (1979: Duluth, Minn.)
Operations research, mathematics and models.
(Proceedings of symposia in applied mathematics; v. 25)
"Lecture notes prepared for the American Mathematical Society Short Course,
Operations Research: Mathematics and Models held in Duluth, Minnesota, August 19—20,
1979."
Includes bibliographies.
1. Operations research. I. Gass, Saul I. II. American Mathematical Society.
III. Title. IV. Series.
T57.6.A47 1979 001.4*24 81-10849
ISBN 0-8218-0029-9 AACR2
ISSN 0160-7634
1980 Mathematics Subject Classification. Primary 90-01.
Copyright © 1981 by the American Mathematical Society.
Printed in the United States of America.
All rights reserved except those granted to the United States Government.
This book may not be reproduced in any form without the permission of the publishers.

CONTENTS
Foreword ix
Mathematical modeling of military conflict situations
by Seth Bonder 1
Queueing networks
by Ralph L. Disney 53
Practical aspects of fishery management modeling
by Fredrick C. Johnson 85
Mathematical modeling of health care delivery systems
by William P. Pierskalla 105
Operations research: Applications in agriculture
by Robert B. Rovinsky 151
Mathematical modeling applied to the relocation of fire companies
by Warren E. Walker 175
vii

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FOREWORD
As part of its educational activities, the American Mathematical Society
sponsors special topic short courses for the attendees of its annual meetings.
This volume contains the revised lecture notes for the short course
"Operations Research: Mathematics and Models " given on August 19-20, 1979 at
the 83rd summer meeting held in Duluth, Minnesota. These lectures emphasized
specific areas of operations research and the mathematics used in modeling and
solving the related problems. The topics and lecturers were:
1. Mathematical Modeling of Military Conflict Situations, Seth Bonder,
Vector Research, Inc.
2. Queueing Networks, Ralph L. Disney, Virginia Polytechnic Institute and
State University.
3. Practical Aspects of Fishery Management Modeling, Frederick C. Johnson,
National Bureau of Standards.
4. Mathematical Modeling of Health Care Delivery Systems, William P.
Pierskalla, University of Pennsylvania.
5. Operations Research: Applications in Agriculture, Robert B. Rovinsky,
U. S. Department of Agriculture.
6. Mathematical Modeling Applied to the Relocation of Fire Companies,
Warren E. Walker, The Rand Corporation.
Each lecturer attempted to make his presentation self-contained in terras of
defining the application areas and mathematics employed. The reader of the
resulting notes will find that the authors, in their desire to broaden the
usefulness of the published material, have, in some instances, stretched the
meaning of self-contained. Thus, the reader might find that a bit of
perseverance, coupled with dipping into some subsidiary references, is required to
obtain the full benefits of the written discussions. However, even the casual
reader will be able to ascertain how the field of operations research has
ix

X
FOREWORD
contributed to the resolution of important decision problems—and how the
field of applied mathematics has flourished in the guise of operations
research.
Grateful appreciation is due all who helped in the running of the short course
and in producing this volume: the authors, Seth Bonder, Ralph L. Disney (co-
chairman of the short course ), Frederick C. Johnson, William P. Pierskalla,
Robert B. Rovinsky and Christine Shoemaker, and Warren E. Walker; the officers
and staff of the American Mathematical Society; the members of the AMS
Committee on Employment and Educational Policy; Alan J. Goldman, who encouraged us
to organize the short course and helped to edit the papers; and to the usual
anonymous referees whose constructive criticisms well served the authors and
editor.
SAUL I. GASS
Editor
University of Maryland
National Bureau of Standards

Proceedings of Symposia in Applied Mathematics
Volume 25, 1981
MATHEMATICAL MODELING OF
MILITARY CONFLICT SITUATIONS
Seth Bonder
Vector Research, Incorporated
ABSTRACT. The resolution of many decision issues (system
developments, force structures, tactics and doctrine, etc.) in the
Department of Defense (DOD) requires information regarding the
results of potential military engagements and campaigns. This
paper will describe some of the mathematical and related modeling
techniques used to generate this information in the tactical (versus
strategic) warfare arena. Background information regarding
historical analytic structures and types of models currently employed in
DOD will be presented. Specific new developments in analytic and
hybrid analytic models in the past 10-15 years will be described and
some numerical results of their use presented.
The birth of operations research (OR) cannot be traced to a specific day
and the roots of OR are as old as science and the management function. However,
its name and the first formal operations research efforts date back to the
defense or military OR activities of World War II. After some 35-40 years it
is my impression that the defense arena is still the most sophisticated and
probably the largest user of OR kinds of analyses. The purpose of this paDer
is to expose you to one dimension of defense OR activities -- the modeling of
tactical military conflict situations.1 Hopefully, this will demonstrate that
lln general, the term "conflict" refers to antagonistic action of one
population on another in order to achieve some goal or objective. This covers a
wide variety of phenomena which include: epidemics, ecological systems
(relations betwen organisms and the environment), prey-predator systems (attrition
of fish populations), military combat between nations or tactical units,
advertising and marketing competition, civil disturbances,
armament/disarmament problems among nations, and social conflict (family feuds). Extensions
and modifications Of the mathematical structures which I shall discuss have
been used in addressing problems in all of these areas.
1980 Mathematics Subject Classification 90B99
Copyright © 1981 American Mathematical Society
1

2
SETH BONDER
mathematical structures can be used to describe operational phenomena in
addition to the classical use in describing physical science phenomena.
I will attempt to accomplish this end with somewhat of an historical
developments perspective as shown in the following outline:
1.0 Introduction
1.1 Defense Planning Issues
1.2 Types of Tactical Warfare Models
2.0 Analytic Model Structures Prior to 1965
2.1 Differential Structures
2.2 Stochastic Duels
2.3 Deficiencies of Existing Analytic Models Prior to 1965
3.0 Model Developments 1965-1975
3.1 The Attrition Rate
3.2 Some Theoretical Results - Homogeneous Forces
3.3 A Battalion Level Hybrid Analytic/Simulation'Engagement Model
4.0 Overview of Current Status and Development Trends
1.0 INTRODUCTION
This section of the caper contains brief descriptions of some
defense planning issues in the tactical (versus strategic) warfare arena and
the types of models that have been developed and used by the defense community
to address them.
1.1 Defense Planning Issues
The forces and systems involved in the tactical warfare area are expensive
to develop, procure, and operate. For this reason, the Department of Defense
performs a large number of studies each year in order to address the broad
spectrum of planning issues noted below:
system characteristics
system choice
system mix
force structure
tactics and doctrine
trade-off among processes
force level
is a requirements issue: to determine the capabilities
that ought to be possessed by next generation systems (e.g., the range and
speed of a new aircraft, detection rates for a new sensor, etc.). The system
choice issue is one of selecting from comparable systems (e.g., the A-10
versus the F-4 aircraft for close air support, the Chrysler versus the
Systems characteristics

MATHEMATICAL MODELING OF MILITARY CONFLICT SITUATIONS
3
General Motors candidate for the new XM-1 tank), while the system mix issue is
one of determining the numbers of noncomparable systems to perform a similar
role (attack helicopter versus close air support aircraft, air defense
artillery versus air interceptors for airdefense). Force structure issues are
concerned with the amounts and relative proportions of the different kinds of
combat units (e.g., number of armored divisions versus the number of infantry
divisions, number of armored divisions versus the number of tactical air wings).
Tactics and doctrinal issues are concerned with choice among alternative
operating procedures for employing the systems and forces. The issue of
trade-offs among processes is in essence one of force structure within a unit
to balance the overall capability of the unit (e.g., the trade-off between
providing resources to acquire targets for fire support and providing fire
support resources to attack the targets, the trade-off between resources for
intelligence collection versus resources for intelligence processing). The
force level issue is concerned with the question, "How much is enough?", and
is, in fact, not a military planning decision but rather a legislative one
involving more political/military and political dimensions.
1.2 Types of Tactical Warfare Models
Over the past twenty-five years a large number of models have been
developed to assist in addressing these types of issues. With few exceptions,
the models have tended to be descriptive (rather than prescriptive) in nature,
describing the battle operations and providing a time history of the status of
the forces throughout the battle. As a broad categorization, three types of
tactical warfare models have been developed and used in the defense area:
war gamesj simulations, and analytic models. Some distinctions between these
models and their characteristics are noted below.
In a war game, command behavior (the decision-making process) is made by
human players, whereas automated logic in the form of engagement or tactical
decision rules or implicit assumptions is used in the simulation and analytic
type models. The modeling (and solution) process of war games and simulations
is distinctly different from that of analytic models. In the former, the
tactical phenomenon of interest is decomposed into its basic activities and
events, and these are organized as they would occur in reality by sequencing
them in time through the use of a network structure or directed graph. Where
required, descriptions of individual event outcomes or individual activities
in the network are developed. Such a model (war gaming, simulation) is then
solved by "acting out" the process in a step-by-step fashion through the
network to generate various battle results (ground controlled, casualties,
etc.).

4
SETH BONDER
In an analogous fashion, the development of an analytic model also
begins with the decomposition and sequencing of events and activities; however,
at this juncture the modeling process differs from that of war games and
simulations. Rather than developing descriptions of each event and activity,
analytic descriptions of event and activity aggregates are developed as
submodels. These submodels are then integrated in a larger overall
mathematical structure. In contrast to acting out the process as a solution procedure,
analytic models are solved by logical mathematical or numerical operations.
In general, any type of model can be either deterministic or stochastic
(probabilistic) in structure. In simplified terms, a deterministic model
always produces the same output for a fixed set of inputs. In contrast,
stochastic models generally require probability distributions on the inputs
and produce probability distributions over the output variables. In practice,
most war games are thought of as being deterministic in nature. In fact, it
is highly likely that, for a fixed set of inputs, varying the players in
different runs of the war game would likely produce different sets of outputs;
however, I am not aware of any war games in which the same situation was
replicated with different players. A large number of the simulation models
used in the tactical warfare area are stochastic and are usually solved by
Monte Carlo sampling methods.1 In essence, for a fixed set of input
distributions the process is acted out via Monte Carlo sampling techniques to produce
a single set of outputs and then this procedure is replicated a number of times
to produce sampling distributions on the output variables. Although it should
come as no surprise to this audience, there are many in the practicing
community who believe that analytic models always must be deterministic in structure.
In fact, analytic models can and have been stochastic in nature: probability
distributions on the inputs generate population probability distributions on
the outputs via standard mathematical solution procedures.
Over the years these three generic model types have been used to develop
models at three levels of tactical engagements: battalion and below
(approximately 40-50 tanks, personnel carriers, attack helicopters, and other
weapon systems versus two to three times this number of opposing weapons),
division/corps level (approximately 25 battalions opposing 75-100 battalions),
and theater level (in Europe, approximately eight NATO corps versus 20-25
Warsaw Pact corps equivalents). The battalion and smaller unit engagement
models are generally used to address some of the more microscopic issues
noted earlier such as system requirements, and system choice. The division
and corps level models are used to address system mix, tactics and doctrine,
and some of the force structure issues. The theater level models are
lSee Wagner (1969).

MATHEMATICAL MODELING OF MILITARY CONFLICT SITUATIONS
5
used to address a number of large-scale force structure issues, logistical
questions, and questions regarding allocation of theater resources to various
locations and missions.
2.0 ANALYTIC MODEL STRUCTURES PRIOR TO 1965
This section of the paper reviews some of the classical
deterministic and probabilistic analytic model structures that were developed prior to
1965 to describe military engagement situations. These were not used
extensively to address defense planning issues at that time due to a number of
inherent deficiencies noted at the end of this section of the lecture.
2.1 Differential Structures
Deterministic Direct-Fire Warfare
Mathematical descriptions of warfare usually attempt to predict the "state"
of the system some time after the"battle is initiated, i.e., the numbers and
locations of surviving forces on both sides. Intuitively, the system state is
a function of the initial numbers of forces, capabilities of the weapon systems,
tactics employed, and characteristics of the operating environment. Most*
formulations assume that it is difficult to hypothesize such a mathematical
function directly. Instead, in many of the deterministic approaches, it is
believed that one can assume something about the rate at which a force is
attritted in a yery small time interval. This assumption regarding the rate
concept leads directly to the use of differential equations as the basic
mathematical structure. Although it is recognized that combat is a random process,
it is commonly considered that solutions of the differential equations are
"expected values."
The classic description of direct-fire warfare activity considers large
numbers of units engaging on a battlefield and hypothesizes that the rate of
attrition is proportional to the number of firing opposing units. Thus,
^- = --m(t) , (1)
and
^- = - Bn(t) , (2)
where
m(t), n(t) = the numbers of surviving Blue and Red forces at time t after
the battle begins,1
xFor notational convenience, the functional dependence of m and n on time
throughout a battle is omitted in succeeding developments, except where its
omission would be confusing to the reader.

6
SETH BONDER
a = the rate at which a single Blue unit defeats Red units (Red
units defeated/time/Blue unit), and
3 = the rate at which a single Red unit defeats Blue units.
a and 3 are usually referred to as attrition rates and are assumed constant over
time, i.e., a,3 ^ f(t), although they may be state dependent. They are factors
that are assumed to account for all other dimensions (weapons, doctrine,
environment, etc.) of combat other than force sizes and are assumed known.
This formulation additionally assumes that
(1) the number of surviving units may be treated as continuous variables,
(2) there are large numbers of homogeneous forces,
(3) these are all exposed and within weapon range of each other,
(4) each unit knows the location of remaining units, and
(5) fire is distributed uniformly over surviving units, i.e., it is
possible to recognize who is killed (perfect intelligence).
We can determine a complete description of the battle by dividing
equation (1) by equation (2) and solving
dn _ oni .
dm " 3n
a(M2 - m2) = 3(N2 - n2) (3)
where m(0) = M, and n(0) = N. It is said that the forces fight to a draw if
m and n approach zero simultaneously, or if
aM2 = 3N2 . (4)
That is, it is considered that the condition
aM2 < 3N2 (5)
is a prediction that the Blue force will be defeated, i.e., winning is defined
as annihilating the opposing force after a long period of time.
Equations (4) and (5) suggest the advantage of concentrating one's forces
in battle. That is, if the Blue system (the weapons, doctrine, environment,
etc.) is four times as effective as the Red system, the latter will need only
twice the initial force size for a draw.
One can easily obtain from equations (1) and (2) the time solutions
m = M cosh /a? t - /jsTa N sinh /a& t,
and
n = N cosh /o|3 t - /a/3 M sinh /a3 t.
Figure la is a graphical representation of the time solution and indicates
annihilation of the Blue force. This result could have been predicted from the
state solution.

MATHEMATICAL MODELING OF MILITARY CONFLICT SITUATIONS
7
aM2 = 9 < 11.6 = 0N2
NUMBER OF
SURVIVORS
OU 1
25 1
20
15
10
5
0
N = 30
M = 20
a = .0236
0 = .0128
-\ ^^^^
1 \^
1 1 1 1 ...l^N. 1 1
0 10 20 30 40 50 60
TIME (SECS)
(a)
70 80
OM = 20 > 11.6 = j8N'
NUMBER OF
SURVIVORS
r
M
a
r
=
ji ii
=
30 '
20
.05
.0128

8
SETH BONDER
(.0236) (20)2 = 9 < 11.6 = .0128 (30)2.
Figure lb shows effect of increasing the Blue force attrition rate to
a = .05.
Deterministic Area Fire Warfare
Thus far, we have discussed the so-called direct-fire warfare equations.
Let me sketch briefly how one varies the functional form of these equations
to suit other conditions. Suppose, instead of being exposed as we assume
here, both sides are completely hidden, and all you know is that they are in
a pea patch over there, and we are in a pea patch over here, and we are
shooting at each other. The terrain is sufficiently obscured so that you can have
any number of troops firing away at each other with neither side necessarily
seeing the target. What happens in this case? Well, once again, let us
assume that we have large numbers of forces on both sides. The rate at which
the N side fires at the M side is proportionate to the number of troops N
has. What is the probability that one man, firing blind, will hit somebody
going to depend on? It will depend upon the density of targets, the number of
troops that are in the area he fires into. The effectiveness of the N element
will depend upon the way he scatters his fire over the area and the relative
amount of that area taken up by his targets, which is proportional to M, the
number of targets. Assuming uniform fire over the area and uniform
redistribution of targets leads to the hypothesis that
$ - - amn, (6)
and
& = - enm. (7)
The state solution with the time variable removed is obtained by dividing
(6) by (7),
= a/3
n) (8)
djl
dm
which leads to
a(M - m)
and an evenly matched battle if1
aM
_ amn
3nm
= 3(N
= 0M.
(9)
1This same result is obtained if one considers the ancient warfare of "Horatio
at the Bridge", The forces are each in single file with only the lead men in
each file engaging with swords, one against one. The numbers at any instant
of time do not influence the battle.

MATHEMATICAL MODELING OF MILITARY CONFLICT SITUATIONS
9
If the Blue force effectiveness per unit is twice that of the Red, the latter
will require twice the number of initial units to effect a balance of fighting
strengths.
There have been many extensions and applications of these basic
deterministic formulations. These include
(a) replacements (arming, when the equations are used to describe
disarmament),
(b) operational losses,
(c) heterogeneous forces
dn.
dt -~ ,l ,i 1
" Z eij*ijn
i
where e..[h..] are the proportions of type i[j] weapons allocated
to fire on type j[i] targets,
(d) guerrilla warfare,
(e) reduction in losses due to retreating, and
(f) reserve commitment.
Guerrilla Warfare
Military operations in Viet Nam (guerrilla-counterguerrilla warfare) have
been modeled by a mix of the linear and square law formulations. The model
considers a defender (m) force moving through an area searching for a guerrilla
(n) force or intending to attack a guerrilla base. The guerrillas counter this
attack by preparing an ambush for the approaching defender. When the battle
starts, the guerrillas fire on the defenders who are in full view. The
defenders' loss rate is thus proportional to the number of firing guerrillas.
Defenders return fire blindly into the area containing the guerrillas.
Accordingly, the guerrilla losses will be proportional to the number of firing
regulars and the number of guerrillas occupying the area. Equations describing
this situation are
j£ = - amn, (10)
g- = - 3n. (11)
It is easy to show that the state solution is
0(N . n) = f (M* - m2), (12)
and the forces will be evenly matched in an engagement of this type when
N = £M2.

10
SETH BONDER
Assuming that this formulation is valid, early data from Viet Nam
1
suggested that «x- < yggg- . We see from the equations that attacking
guerrillas, heavily outnumbered overall, can win if both sides are subdivided
into small groups and the guerrillas always attack with local numerical
superiority. The local superiority required on the part of the guerrilla
is greatly reduced if ambush tactics are used. In this case, the defenders
require yery large local force ratios or extremely effective weapons or both
to win. The attackers (guerrillas) can counter these steps most easily by
reducing their population density in their occupied area during the battle.
If in a local engagement, the ambusher is approximately equal in numbers to
the defender and both have weapons comparable in fighting effectiveness,
then the defender probably cannot win. In practical situations, options
open to both sides to change the outcome will be restricted by the limitations
of resources, environment, and weaponry. Since the defender has much greater
resources available than the guerrilla, the defender's use of guerrilla tactics
would be a powerful tool to defeat the guerrilla force.
Probabilistic Differential Structures
The deterministic models described above miss the details of particular
combats as they v/ould develop in reality, especially for small numbers of forces.
This is because combat is random in nature and develops as a stochastic process
rather than a dynamic trajectory in force-size space. Although not
mathematically correct, solutions of the deterministic equations are usually interpreted
as expected numbers of surviving combatants, which directly imply the existence
of probability distributions of the survivors.
The development of probabilistic models usually assumes that (a) m,n
are treated as discrete random variables, (b) the attrition process is Markov,
and (c) the process possesses a stationary transition mechanism, which may be
intuitively interpreted as follows: the Markov property assumes that, from
any instant of time, the behavior of the system depends on the state of the
system at that instant and not on the previous history of the system. A
stationary transition mechanism assumes that events (losses) which occur in a
given time interval depend only on the state of the system at the beginning of
the interval, and on the length of the interval -- not on the instant at which
the time interval begins.
For illustrative purposes, a simplified stochastic formulation is developed
below which further (but unnecessarily) assumes that in an infinitesimal time
interval, At, (a) the probability of both forces simultaneously losing a unit
is negligible, and (b) the probability of more than one loss on a side is
negligible. Let

MATHEMATICAL MODELING OF MILITARY CONFLICT SITUATIONS
11
P(m, n, t) = probability that there are m, n survivors after a
time interval (0,t),
AAt = probability of one Red casualty in an interval At,
and
BAt = probability of one Blue casualty in an interval At.
Based on the above assumptions, the system can arrive at state (m,n) after the
interval (0, t + At) in three mutually exclusive and collectively exhaustive
ways:
(1) (m,n) survivors at time t, 0 Blue and 0 Red casualties in At,
(2) (m + l,n) survivors at time t, 1 Blue and 0 Red casualties in At, or
(3) (m,n + 1) survivors at time t, 0 Blue and 1 Red casualty in At.
Therefore,
P(m,n, t + At) = P(m,n,t) (1 - AAt - BAt)
+ P(m + 1, n, t) (BAt) (1 - AAt)
+ P(m, n + 1, t) (AAt) (1 - BAt).
Expanding and recombining terms results in the relation
P(m,n,t + At) - P(m,n,t), = _(A + B) P(m>n>t) + B P(m+l,n,t)
At
+A P(m,n+l,t) - A BAt [P(m+l,n,t)
- P(m,n+l,t)].
Taking the limit of this equation as At -> 0 results in
d P(m>n>t) = -(A + B) P(m,n,t) + B P(m+l,n,t) (13)
+ A P(m,n+l,t).
Equation (13), along with
d P(0jt0> t} = A P(0,l,t) + B P(l,0,t),
d Ptm> 0> l) = -B P(m,0,t) + A P(m,l,t) + B P(m+l,0,t)
dt
and
d P(0> n> l) = -A P(0,n,t) + B P(l,n,t) + A P(0,n+l,t),
dt
which are similarly derived, describes the time rate of change of the
probabilities of different states of the system. These recursion relations can be solved
subject to the initial condition that at time t = 0,

12
SETH BONDER
P(M,N,0) = 1, ( m > M (14)
1 or
P(m,n,t) = 0 for ^ n > N .
Using these conditions, equation (13) reduces to
d pty) = .(A + B) P(M,N,t)
which, if A and B are stationary transition mechanisms (constants) gives
P(M,N,t) = e"(A+B)t . (15)
If A or B is a function of time
- S (A+B)ds
P(M,N,t) = e ^ . (16)
We then proceed recursively to find P(M,N-l,t), P(M-l,N,t), P(M-1, N-l,t) ...
with final integration of P(0,0,t).
Having (at least in principle) been able to compute P(m,n,t) it is now
possible to determine the probability distribution for troops on either side
of the battle as a function of time and irrespective of the number of troops
on the other side.
This may be done by computing
N
P^m.t) = ^P(m,n,t), (17)
n=0
= the probability that there are m troops at time
t on the Blue side,
and
M
P2(n,t) = ^2 P(m,n,t), (18)
m=0
= the probability that there are n troops at time
t on the Red side.
One can then compute the average or expected numbers of troops surviving
at time t by
M
m(t) = ^ mP^m.t), (19)
m=0

MATHEMATICAL MODELING OF MILITARY CONFLICT SITUATIONS
13
and
N
n(t) = ]T)nP2(n,t). (20)
n=0
These results may be compared with the results obtained from the
corresponding set of deterministic equations (I remind you that the common
interpretation of the deterministic formulations is that they predict expected
numbers of troops surviving). Generally, they are not the same, and if the
proper comparison is made, usually
m(t) > m(t) and n"(t) > n(t),
although the differences are small for large M, N.
The probability that the Blue side wins (m > 0 with n = 0) is readily
obtained if it is assumed that1
A = ■■ .a
M a + 3
and
B = —£— = 1 - A
a + 3
where a and B are the constant attrition rates defined in the discussion of
deterministic models. In this case, the probability that the Blue side wins
may be determined by viewing the process as a sequence of Bernoulli trials
(casualties) with constant probability of success A that a Red casualty
occurs. The probability of at most M - 1 failures (Blue losses) by the Nth
success (so that Red is annihilated before Blue) is given by
M-l / \ M-l
E(N+i-l\ i A(N+i)-i . V^ (N+i-1)! 1 AN
I N-l ) B A ' 2^J (H-1)! (i)! B A
i=0 > / i=0
which is therefore the probability that the Blue side wins. This, of course,
may be viewed as the probability of crossing the Red force = 0 barrier in a
two dimensional random walk.
Finally, it is important to note that when the stationarity assumption
stated by equation (15) is used, it is equivalent to the assumption of constant
attrition rates employed in the deterministic formulations. That is, although
the transition probabilities are at times considered state dependent, they are
not considered to be functions of time as shown by equation (16) since this
significantly increases the difficulty of analytic solution.
2This is analogous to the deterministic linear formulation.

14
SETH BONDER
2.2 Stochastic Duels
The models discussed so far have been considered macroscopic in that they
consider numbers of forces and their aggregation of weapon effects in the
attrition rates and transition probabilities. The theory of stochastic duels
is considered microscopic because of its concern with microscopic features
such as individual kill probabilities, time between rounds fired, projectile
flight times, etc This is in sharp contrast to differential models which, by
omission, aggregate all of these effects.
We can obtain the flavor of such analysis by considering a simple
"fundamental" duel. It is assumed that this includes
(1) Blue and Red combatants,
(2) fixed single-shot kill probabilities pB and pR, (q=l-p),
(3) unlimited time and ammunition,
(4) projectile flight time = 0, and
(5) firing interval (T) density functions
-xBt
fB(t)dt = Pr(t < T < t + dt) = XBe dt,
mean length of firing interval for Blue,
An
and
-xRt
fR(t)dt = XRe dt.
(6) The firing process is "uncoupled" in that the Blue combatant is
firing at a passive (non-firing) Red combatant and vice versa.
By logical probability arguments, one can develop the pdf for the time for
Blue to fire N rounds T^ = t^ + t2 + ♦.♦ tj. as
N N"1 -V
V e
and
(N-l)!
hB(t)dt = Pr[B kills R in interval (t,t+dt)|B alive]
= PB
(' « "MA / N N-l -XDt\
XRt°e B \ N-l fxR t e B \
—or—y dt +...+ qB pb y-Vrn—J^ +
-PpApt
hB(t) = PBXBe
Analogously
hR(t) = PR xRe
-pRXRt

MATHEMATICAL MODELING OF MILITARY CONFLICT SITUATIONS 15
Then,
Pr[B wins at t] = Pr[B kills R at t|alive]P[B alive at t],
Pr[B wins] = / < hg(t) J hR(t)dt \ dt,
PBXB
pBXB + pRAR
Extensions of stochastic duel theory include
(1) limited duels (ammunition, time, both),
(2) effect of surprise allowing one opponent to fire before the other,
(3) displacement due to near miss causing missed firing turn,
(4) projectile flight time,
(5) triangular duel,
(6) square duel, and
(7) cluster duel (simultaneous firing).
2.3 Deficiencies of Existing Analytic Models Prior to 1965
Many of the analytic combat structures discussed or referred to in
preceding sections were available in the middle 1960s, yet they were rarely
used to address defense planning issues formulated by the advent of the
Planning-Programming-Budgeting System under Secretary of Defense McNamara in
1961. The models were not used, in part, due to a number of inherent
deficiencies, some of which are noted below.
Differential Structures:
no means of predicting attrition rates (transition probabilities)
no consideration of spatial distribution of forces
no tactical maneuver
constant or state dependent attrition rates only
no consideration of terrain or environmental effects
restrictive, matrix exponential solutions to heterogeneous force
formulations
Stochastic Duels:
• difficult to model above one-on-one duel
• omit important parameters
• unrealistic assumptions
• no tactical maneuver
• constant or round dependent parameters only

16
SETH BONDER
Because of these deficiencies, and the belief that analytic type structures
could not represent the complexities of land warfare, Monte Carlo simulations
of small unit (e.g., battalion and below) engagements and war game models of
large scale campaigns were developed and used in the sixties for operational
analyses in defense planning.
3.0 MODEL DEVELOPMENTS 1965-1975
Although Monte Carlo simulation and war game models of land warfare were
developed and used to assist defense planning in the sixties (and still are),
it was recognized that more efficient structures would be needed to perform
studies that were responsive to annual defense planning cycles. War game
models at times took six to eight years to develop and, as late as 1972, one
of the better division level games took six months to simulate ten hours of
combat in a study. For a fixed battle situation, one of the more sophisticated
battalion-level simulations required 30-60 minutes per replication on an
IBM 370-95 computer. (Needless to say, Monte Carlo simulation approaches have
not been considered to represent division, corps, or theater level battles.)
Accordingly significant, albeit uncoordinated, research and development
activities were conducted in the 1965-1975 time period to improve the inventory
of combat models for defense planning. Some of these research and development
areas are noted below.
Prediction of attrition rates (transition probabilities)
• different weapon systems
• different firing doctrines
• different types of engagements
Spatial distribution of forces
Effect of tactical maneuver
Terrain and environmental effects
Heterogeneous force formulations
• analytic
• hybrid analytic/simulation
independent
• free parameter
Structured programming
Efficient solution procedures
Large scale unit (division, corps, theater) modeling
Strategy research
• optimal allocation of effort
• optimal tactical maneuver
• optimal weapon system mix

MATHEMATICAL MODELING OF MILITARY CONFLICT SITUATIONS
17
Clearly, it would be infeasible in this paper (and probably not instructive
given its objective) to touch on each of the research and development efforts.
Instead I will (1) describe in section 3.1 the development of one of the
attrition rate models, (2) briefly display in section 3.2 some results of
theoretical work on the effects of maneuver and concomitant time variations in the
attrition rate, and (3) in section 3.3 describe the structure of one type of
battalion-level hybrid analytic/simulation model employed heavily in defense
planning. Finally, in section 4.0 I will summarize the current status of combat
model developments at each of the force levels (battalion, division/corps,
theater) and note some emerging trends.
3.1 The Attrition Rate
Concept of the Attrition Rate
The attrition rate for individual weapon systems was assumed to be
dependent on a multitude of physical parameters of a weapon system which describe
its capabilities in such areas as acquisition, firing accuracy, delivery rate,
and warhead lethality. Experience with existing systems suggested that these
characteristics are dependent on the range to a target and are stochastic in
nature. That is, the attrition rate is functionally dependent on the range
between combatants and, for any specified range, is described by a probability
distribution. In the vernacular of the mathematician, the attrition rate may
be viewed as a nonstationary stochastic process when forces employ mobile
weapons. This is shown in figure 2, which depicts the two distinct variations
in the attrition rate for a single weapon system type against one target type:
(a) the stochastic variation at a specific range, which is described by the
conditional probability distribution f(a|r), and (b) the variation in some
function of the attrition-rate random variable with range, which is called the
attrition-rate function, a(r).1
The fact that armed conflict is stochastic is well recognized and is one
of the reasons for conceptualizing the attrition rate itself as a nonstationary
stochastic process, P[a,r], Assuming the process P[a,r] could be predicted,
one would like to incorporate the range and chance variations of the attrition
rate explicitly into a model of combat among heterogeneous forces. The rate
concept suggested that such a model would be either a differential equation
(continuous-state variables) or a difference-differential equation (discrete-state
variables) structure in which the relevant coefficients were nonstationary
xFor clarity of discussion, variations in the attrition rate due to changes in
target posture, environmental effect, etc., which can be included in the model,
are not presented in this paper.

18
SETH BONDER
Range r
FIGURE 2 THE ATTRITION-RATE PROCESS
stochastic processes, i.e., the P[a..,r] and P[3..,r] for all weapon-target
group pairs. Initial study strongly indicated that, in the foreseeable
future, there was little hope of solving either of these structures even for
simplified situations. A research decision was made to suppress the chance
variation in the attrition rate and to concentrate on structures of combat
which explicitly involved the range variation in the rate when mobile weapons
are employed.
Discrete-state stochastic process models were considered in which the
transition rates are nonstationary, i.e., as varying with time. The literature
indicated that discrete-state stochastic process formulations of combat have been
difficult to solve even when the process is considered to be Poisson with*
stationary transition mechanisms. The few solutions obtained with homogeneous
forces had been of such complexity as to delimit their usefulness for analysis
purposes (Dolansky, 1964; Clark, 1968). Accordingly, it was felt that useful
solutions for general discrete-state stochastic process formulations with non-
stationary transition mechanisms could not be obtained in the near future.
Although the appropriate long-range objective is to develop stochastic
formulations of heterogeneous-force armed combat such as those noted above, it
was felt that a more reasonable intermediate objective would be the development
of deterministic formulations, and solutions, which included the non-stationary
aspects of the attrition rate at the expense of explicit considerations of its

MATHEMATICAL MODELING OF MILITARY CONFLICT SITUATIONS
19
stochastic elements. Accordingly, the coupled sets of differential equations
$ - -«(r)m (21)
$ = -6(r)n (22)
were chosen as the mathematical structure to model the combat activity between
homogeneous forces. The nonstationary aspect of the attrition rates is
included in the formulation as the variable coefficients in the differential
equations, where the variable coefficients are appropriately defined as the
attrition-rate functions, a(r) and a(r).1 Thus, there is one value of the
attrition rate (for any firing weaDon on a specific target group) at each range.
Definition of the Average Attrition Rate
The attrition rate at each range was defined to be the harmonic mean of
the attrition-rate random variable. The appropriateness of this definition for
use in a differential equation model of combat is seen below.
Consider a homogeneous-force battle in which the initial numbers of
Blue (M) and Red (N) forces are sufficiently large so that neither is
totally annihilated. Each Blue weapon system is engaged in a renewal process
of attriting targets, i.e., the times between kills are independent and
identically distributed random variables0 From Blackwell's theorem (Parzen, 1962,
p. 183),
Lim Pr[renewal in (t, t + dt)] = — ,
t-H» u
where
y = the expected interrenewal time.
Therefore, the expected number of Red kills in (t, t + dt) is
E[number of Red kills in (t, t + dt)] = — . (23)
The differential equation homogeneous-force model of combat (equation 21)
states that
-dn = E[number of Red kills in (t, t + dt)] (24)
= amdt.
Comparison of (23) and (24) suggests that a be defined as 1/y. More
generally, the definition of the attrition rate to use (for a specific range)
in the differential equation structure of heterogeneous-force combat is
Assuming a one-to-one correspondence between range r and time t.

20
SETH BONDER
uei -I
aij(at range r) = ETrTTTI • (25)
where
E[T..|r] = the expected time for a single Blue system
of the i group to destroy a passive j
group Red target, given it is firing at a
target at range r.
This definition also leads naturally to defining the range variation of the
attrition rate as the variation in the reciprocal of E[T..|r] as the
range to the target changes. The range variation is called the attrition-
rate function and is denoted by a..(r), as used in the differential equation
structure of combat.
Attrition Rate for Tank Systems
In this subsection we develop an attrition-rate model for a tank gun
system which is referred to as a "repeated single-shot, Markov fire" weapon
system. The resultant model is the probability density function and the expected
value for the tine-to-kill random variable against a specific type target at a
specific range since, by definition, it is used to predict the attrition rate.
A straightforward analysis of the physical process will be used to develop the
model to illustrate modeling techniques, even though more elegant mathematics
such as transform techniques and Markov-renewal processes1 can be employed.
Implicit in this type of development are several assumptions which are listed
here as a convenient summary and reference. These are
(a) the systems are of the impact-lethality, repeated single-shot,
Markov-fire class,
(b) the probability of kill given an impact is identical for eyery
round fired,
(c) the time preceding the firing of the first round is not
random, and the conditional times to fire a second round after
a hit and after a miss are not random,2
(d) the probability that a round fired after a preceding hit or
miss results in a hit or miss is not influenced by the knowledge
of other history of the engagement (such as the number of rounds
fired or the number of previous hits),
(e) the engagement terminates immediately on a kill.
lSee Bonder and Farrell (1970, Part B, Chapters 3 and 4).
2These assumptions can be relaxed using the transform and Markov-renewal
techniques.

MATHEMATICAL MODELING OF MILITARY CONFLICT SITUATIONS
21
The firing doctrine for a main tank gun varies from round to round as shown
in figure 3. Figure 3(a) shows the adjustment procedure following a hit on
the first round which is to replace the crosshairs on the target—presumably
the position of the crosshairs for the first round. Figure 3(b) depicts the
"burst-on-target" adjustment doctrine following a miss on the first round.
Succeeding adjustments, based on the result of the immediately preceding round,
are made in a similar fashion until the target is defeated. The probability
density function (pdf) of the time to accomplish the result is obtained by
essentially modeling this adjustment process as it occurs, round by round.
Since the objective of a weapon system is to defeat the enemy, we begin
by defining lethality and its unit of measurement. In brief, lethality refers
to what happens to the target when struck by a projectile. The particular
effect of interest is the target's combat utility. When this combat utility is
reduced to zero, the target no longer poses an active tactical threat and may be
considered defeated or killed. The definition of a defeated or killed target
is, of course, dependent on the target's mission or role in combat. For example,
consider an armored tank which is frequently referred to as "mobile, protected
firepower." Some of the tank's combat missions require primarily firepower,
others require mobility, and still others require both firepower and mobility,
and the definition of lethality must consider which of these are relevant in
the context of a study.
Lethality against a particular target is measured as the conditional
probability of a kill, given the projectile hits the point target, and noted
symbolically as either P(K|H) or P„. This measure is dependent on the
mechanical damage caused by perforating and/or striking the target, and the loss
in combat utility resulting from this mechanical damage. Procedures to predict
this measure for different types of targets have been developed. See, for
example, Zeller (1961), Goulet (1963), Freedman (1965), and Meyer (1967).
Another measure of lethality can be defined as "the number of hits, z,
needed to defeat the target." Since we are concerned with destroying the target
just once, this measure is directly related to the conditional kill probability
by the geometric density function
p(z) = (1 - PK)Z"\ • (26)
The number of hits needed to defeat the target, z, is initially used as a
parameter in subsequent developments of this section.
The number of hits required to effect a kill describes a weapon's lethality
characteristics against particular targets. The weapon's accuracy capabilities
are next considered by developing the distribution for the number of rounds fired
(hits and misses) to defeat the target.

22
SETH BONDER
Crosshairs
Projectile
Impact on
Target
Crosshairs
(1) First-Round
Aim Point
(2) First-Round Hit
(3) Second-Round
Aim Point
(a) Relay Following a Hit
Crosshairs
Projectile Burst on
/ Burst /Target
Crosshairs
(1) First-Round
Aim Point
(2) First-Round Miss (3) Second-Round Aim
Point (Burst on Target]
(b) Adjustment Following a Miss
FIGURE 3 FIRING DOCTRINE FOR A MAIN TANK GUN

MATHEMATICAL MODELING OF MILITARY CONFLICT SITUATIONS
23
Let
P, = first round hit probability,
p = conditional probability of a hit given the preceding
round fired missed the target,
u = conditional probability of a hit given the preceding
round fired hit the target,
and consider the sequence of trials (rounds fired) connected in a regular
Markov chain with transition probability matrix
P] hit
(1 - P,) miss
It is assumed that p and u are defined only on the open interval (0,1).
We seek the pdf for the number of rounds, N, to obtain z hits if the
sequence of firings ends with a hit.1 This can occur in two mutually exclusive
and collectively exhaustive ways.
f(N|z) = f(N-H-H|z) + f(N-M-H|z) . (27)
The first term on the right-hand side of (27) is the probability that the
first and last rounds of the sequence result in hits given that the z hits
occur in N firings. The second term is the probability that the first and
last rounds of the sequence result in a miss and a hit, respectively, given
that the z hits occur in N firings.
To determine f(N«H-H|z) we consider the following combination of firing
results:
In the first r, firings, the event hit occurs every time;
In the next s, firings, the event miss occurs every time;
In the next r~ firings, the event hit occurs e\/ery time;
In the next s9 firings, the event miss occurs e\/ery time;
In the next s. , firings, the event miss occurs every time;
In the last r, firings, the event hit occurs every time.
xThe procedure could be extended to remove this assumption that the firer
recognizes when the target is defeated without technical difficulty but
with increased complexity of discussion.
hit miss
u 1 - u
1 - p
0 < u < 1
0 < p < 1

24
SETH BONDER
The joint occurrence of these events has the probability
r,-l srl r?-l s?-l r.-l
P]U ' (1 - u)(l - p) ' pu d (1 - u)(l - p) Z p...pu k
rl+r2+- • -rk"k k l s^+Sp.-.s. r(k-l) . ,
= ^u ' Z k (1 - u)k '(1 - p) ' 2 k~' pk"'-
(28)
Since there are a total of z hits and (N - z) misses,
k
/ A r. = z and > s_. = N-z
i=l
Therefore, (28) becomes
k-1
i=l
Pluz"k(l -uj^d - pj^-^p"-1
Accordingly, the probability of the outcome depends only on N, z, and
k and not on the values of r. and s.. The number of hits, z, can be
expressed as a sum of k positive integers (the r
,><» (a)
ways and the
number of misses, (N-z), as a sum of (k-1) positive integers (the s.) in
I "k\ )ways- Therefore, the probability that it takes N firings to obtain
z hits, the first and last being hits with probability P, and p or u,
respectively—where the hits occur in k groups and the misses in (k-1)
groups--is
(*:!)("£') «'
'd - ury-'o - pi'
The outcome can occur for all values of k such that (1 < k < z)
Accordingly,
r?y
N = z
f(N-H-H|z) =<
since
(w)
k=2 v '
z"k(l - u)k"1pk-1
("£')
(1 - P)
N-z-k+1
N > z
(29)
= 0 when k = 1 and N > z.

MATHEMATICAL MODELING OF MILITARY CONFLICT SITUATIONS
25
By an analogous derivation, it can be shown that
z
£ (-)"!
k=l > '
f(N.M.H|z) = (1 - Pn) > ;[^)uz-k(l - u)k_1pk
(X)
(1 - p)
N-z-k
for N > z .
(30)
Substituting (29) and (30) into (27) completes the derivation for
N = z
f(N|z) =/
htfcO"1-'"-■>'-'''("ft1)
N-z-k+1
.N-z-k
(31)
N > z ,
where Q-, = (1 - P,) and q = (1 - p). The reader is reminded that equation
(31) is a conditional distribution which is dependent on the integer z.
It is a straightforward matter to show that the characteristic function of
(31) is
♦N.z(s) - E[eisN] = £eisNf(N|z)
N=0
= e
1SZ
P] + ve
IS
1 - qe
is
u + (] - u)Pe
is
1 - qe
is
z-1
(32)
where s is a dummy variable and i Setting s = 0 in (32),
00
,|z(o) = 2f(N|z) = ] '
"N
N=o
proves that (31) is, in fact, a probability density function. The expected
value of N is obtained from (32) as

26
SETH BONDER
E[N|z]
1 d*N[z(s)
i ds
- r , ° " P1} , (1 - u)(z - 1) . (33)
P P
The density function f(N|z) for the number of rounds that must be
fired to destroy a particular target is dependent on the lethality and
accuracy capabilities of the weapon system. Two other important weapon
characteristics remain to be considered -- the system's acquisition capabilities
and its rate of fire. We consider these characteristics in a manner such that
the acquisition and firing processes are serial. That is, targets are destroyed
by sequentially acquiring a target, attriting it by fire, acquiring a new
target, attriting it, acquiring a new target, etc. This is in contrast to
parallel acquisition and firing processes in which new targets may be acquired
while a previously acquired one is being attrited.
We include the timing characteristics of acquisition and firing by
defining
t = the time to acquire targets,
a
t, = time to fire the first round,
t. = time to fire a round given the preceding round was a hit,
t = time to fire a round given the preceding round was a miss,
xf = projectile flight time,
and consider the following sequence of events from target acquisition to
destruction. The sequence begins with detection which takes t time units
a
to occur. The first round is then fired and arrives at the target area
(t-, + xf) time units later. If the first round misses, the next round will
arrive (t + xf) time units after the first. If the first round hits the
target, and more than one hit is required (z > 1), the next round will
arrive (t. + xf) time units later. The sequence of firing after hits and
misses is continued until the final hit which destroys the target is obtained.
This description is consistent with our single-shot Markov firing doctrine in
which the result of the previous round is observed before the next one is fired.

MATHEMATICAL MODELING OF MILITARY CONFLICT SITUATIONS
27
In this process, rounds will be fired after each of (z - 1) hits and
(N - z) misses. Accordingly, the time to defeat a target may be written as
T = Ta + (t1 + Tf} + (xh + Tf)(z - 1} + (xm + Tf)(N " z)
c1 + c2N ,
where
T^ + Tn - T. + (l. - Tm)z
a l h n m'
2 m f
(34)
(35)
(36)
Equation (34) defines T as a linear function of the discrete random variable
N, and establishes a one-to-one transformation between their respective sample
spaces. The density function of T is readily obtained from (31) by the
change of variables technique for discrete variables as
f(T|z) =/
T = c, + c~z
k=2
+ Qn
I
£(!:!)■
k=l
z-k+1
■z-k
T > c, + CpZ
(37)
The characteristic function of T, 4>Tiz(s), is obtained directly from (32) and
the definition of T given in (34), as
E [>]
♦TU(s)
= e
is(c]+c2z)
Q^e
ic«s
Pn + - ic9s
' (1 - qe l )
1 z-1
u + (] - ^)Pe
1C2S
1 - qe
TCpS
(38)
The expected value of T can be obtained from (38), or, more directly, by
employing the linear property of the expected-value operator with (34).
Accordingly,

28
SETH BONDER
E[T|z] = c] + c2E[N|z]
♦^
"'V + U-DO ~u) + z] .
'1 z\ p p
The expected time to destroy a target, E[T|z], is conditioned on the
(39)
integer-valued lethality variable z, which is the number of hits required
to destroy the target. This conditioning is removed and the continuous
lethality parameter P^ (the conditional probability of destroying the target
given it is hit by a projectile) introduced by
E[T] = ^E[T|z]p(z)
z=l
= T +
fTh + Tf>
r^m + V
Ll
lh
1 - u
+ u - P
, (40)
where p(z) is given by equation (26). Finally, the attrition rate for tank
gun fire is given by
def _J_
E[T] (41)
Attrition rate models for many different types of weapon systems and firing
doctrines have been developed.
3.2 Some Theoretical Results - Homogeneous Forces
Given the availability of attrition rate models as a function of basic
weapon system performance capabilities (e.g., accuracy, acquisition, firing
times, lethality, etc.) and information on how these capabilities change with
range to a target, the attrition rates as a function of range, a(r) and
3(r), can be calculated directly. These are used as data in a number of
models used in defense planning studies such as the one which will be
described in section 3.3. Additionally, a significant amount of
theoretical work has been performed with mathematical forms of a(r) and
3(r) in conjunction with equations (21) and (22). Thus, converting equations
(21) and (22) into an appropriately defined spatial dimension and combining,
produces the equation
^
n f w
1 dct(r)
a(r) dr
le-m*
= 0
(42)

MATHEMATICAL MODELING OF MILITARY CONFLICT SITUATIONS
29
describing the number of surviving "Red" forces (n) as a function of the
distance between forces, where
r = the range between forces
v = relative velocity between forces
w = relative acceleration between forces.
A similar equation for the surviving Blue forces (m) exists.
Solutions of equation (42) and the analogous one for the Blue forces
for different attrition rate functions, a(r) and 3(r), have provided some
insights regarding combat dynamics. For example, it is believed that the
Soviets are willing to accept heavy tank losses in using velocity and mass
to achieve a successful attack. Figure 4, derived from equation (42) for
particular a(r) and a(r), suggests that use of velocity and mass may be
the best way to reduce losses if a force wishes to attack. The curves suggest
that increasing the attack velocity increases the number of survivors but with
decreasing marginal returns. For a fixed attack velocity, higher initial force
ratios conserve attackers. In essence, speed and force concentration are good
ways to saturate a defender's retaliatory capability. Some additional
theoretical analyses of tactical maneuver spatial dynamics are presented in Bonder and
Farrell (1970).
SURVIVING
NUMBER
OF
ATTACKERS
AT
OBJECTIVE
2.5 INITIAL
FORCE
RATIO
1.67
(ATTACKER/
1.25 DEFENDER)
1.0
ATTACK SPEED
FIGURE 4 SURVIVING NUMBER OF ATTACKERS AS A FUNCTION OF ATTACK SPEED

30 SETH BONDER
3.3 A Battalion Level Hybrid Analytic/Simulation Engagement Model
The ability to predict attrition rates as a function of measurable (or
predictable) weapon system capabilities has facilitated the development of a
number of hybrid analytic/simulation models of military battles which,
because of their responsiveness, have supplemented (and oftentimes replaced)
Monte Carlo simulations and war games as vehicles for defense planning studies.
A hybrid analytic/simulation model is a model which combines analytic and
simulatory approaches to modeling the constituent processes of combat, usually
employing numerical solution techniques. This section of the paper
describes the structure of one of the earlier battalion level hybrid models
developed for the Army in 1969. This model, and its numerous sucessors, have
been used extensively to address many DOD requirements and system choice issues.
In this model, the attrition, target acquisition, and intelligence processes
are modeled analytically, while the movement, command and control (decision),
and terrain effects processes are simulated.
The model is a deterministic differential one which focuses on small
time intervals during the battle. In particular, for each side, it is
hypothesized that in a short period of time
(a) locations change due to tactical movement,
(b) weapon systems are attrited by enemy activity,
(c) resources are expended, and
(d) personnel become casualties due to enemy activity.
For purposes of this discussion, I have neglected the possibilities of arrivals
and resupply during the small interval of time; they can readily be included.
As with the differential models described earlier, it is assumed that, if
we know the state of the battle at the beginning of the small interval, we can
predict the rate at which weapons systems and personnel are attrited during this
small interval. For convenience, we assign names to the numbers of different
groups of systems in each force. Let
m. = the number of surviving Blue units of the i group (i = 1,
2, ..., I),
n. = the number of surviving Red units of the j group (j = 1,
2, ..., J).
Different groups are determined by their ability to attrit weapons systems of
an opposing group or be attritted. Therefore, a missile weapon system and a
rapid-fire machine gun form different groups since the rates at which they can
attrit targets of an opposing group are different. Additionally, similar weapon
system types can form different--groups if they are at different ranges to the
target and this range difference affects their ability to attrit it. Thus, a
tank platoon at 1,000 meters to the target forms a different group than another
tank platoon at 2,000 meters from it.

MATHEMATICAL MODELING OF MILITARY CONFLICT SITUATIONS 31
We assume that
(a) the rate of loss of units in the j Red group due to the i
Blue group is proportional to the number of units in the i
Blue group with a proportionality factor called the attrition
coefficient, and
(b) the rate of loss of units in the j Red group in total is the
sum of the rates of losses due to different i Blue groups.
Mathematically, these assumptions take the form of the following coupled sets
of variable-coefficient differential equations to describe heterogeneous-
force battles:1
dt
dm.
~dT
"S Aij(rjjK for j = ]> 2> •••' J> (43)
i
"S Bji(rij)nj for i = 1, 2, ..., I, (44)
where
A..(r..) = the utilized per system effectiveness of systems
'J 'J th tn
in the i Blue group against the j Red target
group at range r. This is called the Blue
attrition coefficient.
B..(r..) = the utilized per system effectiveness of systems
J1 "U th th
in the j Red group against the i Blue target
group at range r. This is called the Red
attrition coefficient.
Although the variable r.. is used to designate the range between the firing
weapon group and the target group, it should be noted that, in application of
the model, actual time trajectories and positions of each group are considered.
Additionally, although not explicitly shown, resources expended are included
in the development of the A.., and can be determined directly from the model.
It is noted that this formulation is a deterministic one which treats the
numbers of surviving forces (m. and n.) as continuous variables, while
clearly the actual battle activity is a random phenomenon and m. and n. are
integer-valued variables. Although many probabilistic arguments are contained
Battles in which at least one of the forces has more than one group.

32
SETH BONDER
in this formulation, the output of the model is a deterministic trajectory of
the surviving numbers of forces.1
The attrition coefficients (A., and B..) are, as one would expect,
complex functions of the weapon capabilities, target characteristics,
distribution of the targets, allocation procedures for assigning weapons to targets,
intelligence, etc. The model attempts to reflect these complexities by
partitioning the total attrition process into four distinct ones:
(1) the effectiveness of weapons systems while firing on live
targets,
(2) the allocation procedure of assigning weapons to targets,
(3) the inefficiency of fire when other than live targets are
engaged, and
(4) the effect of terrain on limiting the firing activity due to
loss of acquisition capability and also on mobility of the systems.
These effects are included in the attrition coefficient as
Aij(r..) = ^(^.(^..(r^F^t) (45)
W = eji^i>hii(ri}VriJ)GJi(t) (46)
where
a,-,-(r) = the attrition rate. The rate at which an individual
system in the i Blue group destroys live j -group
Red targets at range r.. when it is firing at them.
th
e--(r) = the allocation factor. The proportion of the i Blue
'J th
group systems assigned to fire on the j -group Red
targets which are at range r...
^ th
I..(r) = the intelligence factor. The proportion of the i
1J th
group firing Blue weapons allocated to the j Red group
which are actually engaging live j -group Red targets
at range r...
th
F-.(t) = the terrain/acquisition factor. The oercentage of i group
J th
survivors who have detected a tarqet in the j target group
at time t.
Similar definitions exist for the components of the Red attrition coefficient,
B... In section 3.1 of this paper we defined the attrition rate at a
Research done on comparing the deterministic and stochastic formulations for
the homogeneous-force case (only one force group on each side) indicates that
the deterministic formulations are reasonably good approximations to the
expected number of survivors if there is a small orobability that either side
is annihilated. Additionally, it is noted that in many defense studies that
employ Monte Carlo simulations, only the expected results are considered in
the decision-making process.

MATHEMATICAL MODELING OF MILITARY CONFLICT SITUATIONS
33
particular range1 to target as
and described a model for predicting the rate for a tank gun system as a function
of its various performance capabilities. Similar models have been developed for
other types of weapon systems and firing doctrines. Methods developed to predict
the other components of the model are described in succeeding subsections.
The Allocation Factor
As noted earlier, the allocation factor is the proportion of the i Blue
group systems assigned to fire on j -group Red targets. This is included since
only those systems directing their fire (or other lethal effects) on the j
group or its area are likely to cause attrition of the target. The allocation
factor may be input by military judgment reflecting the assignment strategies
deemed most appropriate to the tactical situation. This factor may be input
directly or determined from a priority or target worth scheme.
In addition to the use of military judgment to assign weapon groups to
target groups (the procedure usually employed in the model when used by the
Army), research results on allocation strategies using differential game
concepts (Isaacs, 1965) have given rise to an approximate optimal2 priority
ordering rule which is used in some simpler versions of the model. The rule is
"Blue weapon group i should engage live Red target group K
if the product
aiK6Ki > aij3j for all j
and Red weapon group j should engage live Blue target group
K if the product
BjKnKj > Vij for a11 1'"
This rule is used over all eligible targets, which consist of those targets
within range which are not externally prohibited.3 If all eligible target
groups are unable to return fire, then all of the above products will be zero.
xFor clarity of discussion, variations in the attrition rate due to changes in
target posture, environmental effect, etc. which are included in the model are
not presented in this section.
20ptimal with respect to linear end of battle measures such as the difference
(Blue minus Red) in survivors.
3Externally prohibited targets for a weapon group are those for which the
attrition rate is zero, e.g., a rifle against a tank target.

34
SETH BONDER
In this case, Blue group i is assigned to fire on that target group j for
which a-, is maximum and an analogous rule is used for assignment of a Red
weapon group.
The Intelligence Factor
As noted earlier, the intelligence factor is the proportion of the i
group firing Blue weapons allocated to the j Red group which are actually
engaging live j -group Red targets. This factor is included to consider the
loss in efficiency (effectiveness) of a firing weapon when it is firing on
either targets already attrited or on areas that are void of targets. Based on
some renewal theory modeling, the intelligence factor is predicted as
I,,(r,.J = -^ , (47)
where
PLTL + PDTD + PVTV
p. = the probability of firing on a live target, given fire on a target,
PD = the probability of firing on a dead target, given fire on a target,
Pv = the probability of firing on a void area, given fire on a believed
target,
T. = the expected or average time to fire on a live target before
switching fire (given by equation 40),
TD = the expected or average time to fire on a dead target before
switching fire, and
Ty = the expected or average time to fire on a void area before
switching fire.
Fire is directed at attritted targets or voids (false targets) because of the
uncertainties associated with detecting, recognizing, and identifying real
targets under the stress of battle. The intelligence factor dependence on range
is due to both the probabilities and the times in equation (47).
Computation and Terrain Interactions
As implied throughout this paper, there exist a number of operating versions
of the differential ground combat models varying in degrees of complexity and
simplifying assumptions. At one end of the continuum are simplified models
(homogeneous forces, constant attrition-coefficient heterogeneous forces, no
terrain effects, etc.) which succumb to closed-form mathematical solution
techniques. These are used principally for theoretical research. At the other end
of the continuum are more realistic complex versions that include more terrain

MATHEMATICAL MODELING OF MILITARY CONFLICT SITUATIONS
35
effects and larger dimensionality in the input data (i.e., kill probabilities,
which depend on the target type, cover status, movement status, aspect angle,
etc.). These models usually require the use of numerical solution
procedures and are used principally for analysis in weapon system studies.
Prior to discussing the terrain/acquisition factor, I shall briefly summarize
the computational procedure employed and the way that terrain effects are
included in one of the versions of the model.
In the computational program, the basic differential equations are
approximated by the difference equations
I J
0, m.(t) - Y^ B..(t)n.(t)At
j = l
for i = 1, 2, ..., I
II
°> M*) " y^J Ai j(t)m.(t)At
i=l
for j = 1, 2, ..., J,
where At is the computational time step. A 10-second time step is usually used.
In applications of the model to battalion-level task force engagements with
weapon systems currently under study it was observed that smaller time steps
did not alter the solution, while larger steps led to significant errors
(overkills, failure to switch assignments, etc.). Clearly, the time step must
be appropriately selected depending on the capabilities of the systems involved
and the scenario activity.
The correspondence between battle time and the spatial distribution of
forces during the battle is obtained from knowledge of predetermined movement
patterns of all Red and Blue groups which are input to the model. These
movement patterns are obtained from a terrain preprocessor. Routes of advance and
movement tactics (sections leapfrog, sections advance and provide covering fire,
etc.) are input to the preprocessor, which then considers the terrain
characteristics (soil type, roughness, grade, etc.) along the routes and maneuver
capabilities of the weapon systems (speeds, accelerations, etc.) to generate the
time-sequenced movement patterns.
The effects of terrain which limit the firing activity due to loss of
acquisition capability are also included in this model. For this purpose the
terrain is incorporated in this version of the model as if it were a map with
digitized properties of concealment, cover (line-of-sight), etc. associated
with each location or pairs of locations. The model considers these terrain
effects (which are also obtained from the preprocessor) in developing the
terrain/acquisition factor.

36
SETH BONDER
Terrain/Acquisition Factor
The terrain/acquisition model considers that the acquisition process
occurs in parallel with the firing and movement processes. Since the overall
model considers groups of weapon systems, the parallel acquisition submodel is
designed to determine the percentage of observers in a group who have detected
a target in an opposing group. Detections occur due either to visual sighting
of nonfiring targets or pinpointing the flash of a firing target. The terrain
effects of visibility (fully exposed, partially exposed, not exposed) and
line-of-sight1 (exists, does not exist) are interfaced with the visual
detection and pinpointing capabilities by the following computational formulas:
F^t+At)
percentage of i -group survivors who have
failed to detect a target in j target group
at time (t + At)
1
if no LOS
(48)
Fij(t)Vij(t, t + At)Pij(t, t + At)
if LOS exists,
where
,th
ij
V^(t, t + At) = percentage of i "-group survivors who have
failed to visually detect a target in j target
group in the interval (t, t + At)
1
Pij(t'
t + At)
if visibility does not
exist
-Ai .(t+At)-At-n.(t)
.th
(49)
if visibility exists
percentage of i""-group survivors who have failed
to pinpoint a target in j target group in the
interval (t, t + At)
and
xij(t)
nj(t)
1
(1
P..(t,t+At)
"\ (t) n.(t)
if no LOS
if LOS exists
(50)
= the detection rate at time t, which is a function
,th
.th
of range between the i and j groups and exposure;
the number of surviving j -group targets at time t;
Abbreviated LOS.

MATHEMATICAL MODELING OF MILITARY CONFLICT SITUATIONS
37
p.. = probability of i -type observer pinpointing a
th
j -type weapon when it fires one round. This
is considered dependent on range between i and
j groups, the weapon type, observer type, and
movement status;
p..(t, t + At) = the number of rounds fired by the j target group
in the interval (t, t + At).
The formulas assume a Poisson visual detection process when visibility exists
and a geometric pinpointing process when LOS exists.
The percentage of i -group survivors who have detected a target in the
j target group at time (t + At) is obtained directly from F..(t + At) for
all j. Using this information, and an input military worth or target priority
rule, appropriate numbers of the i -group survivors are assigned to fire on
different target groups.
Model Output
Output of this version of the model at each time step is a status listing
of all weapon groups which are involved in firing events (firer or target).
The listing contains the group numbers, their locations, range separations,
movement status, visibility status, percentage of the firing group allocated,
round-type used, attrition rate, and amount of attrition. A summary output is
also provided each 10-second time step. This summary lists the cumulative
number of losses of each weapon type by the types in the opposing force which
caused the attrition.
Comparisons with Monte Carlo Simulations
When initially developed, the defense community was \/ery reluctant to use
this hybrid analytic/simulation model because of its belief that only detailed
Monte Carlo simulations with their focus on individual systems and activities
could credibly represent the complexity of a land battle. For this reason the
model was "tested" by comparing its combat predictions to the results predicted
by more detailed Monte Carlo simulations. A few of the comparison results are
presented in this subsection of the notes.
Figure 5 depicts one of the tactical plans considered in one of the
comparison studies. The tactical plan shown is a Blue attack engagement
against a fixed Red defensive position. The attack is conducted along three
major axes with four individual routes of advance per axis. Each route
consists of individual main battle tanks and/or supporting armored personnel

TACTICAL PLAN NO. 2
Battle Tank
bd Personnel
Groups (for each
)
|e Long-Range
iles
A ATTACK POSITION
O OBSTACLE
e AXIS OBJECTIVE POINT
© IUA OBJECTIVE POINT
CD ROUTE OBJECTIVE PCINT\
© REORGANIZATION POINT
O ROUTE
• ROUTE DESCRIPTOR
FIGURE 5 TACTICAL PLAN FOR BLUE ATTACK ENGAGEMENT

MATHEMATICAL MODELING OF MILITARY CONFLICT SITUATIONS 39
carriers equipped with rapid-fire weapon systems. In addition to these maneuver
units of main battle tanks and personnel carriers, the Blue attack force had
long-range missiles and short-range missiles, as shown in the figure. The
defending force is comprised of tanks, missiles, and armored personnel carriers
equipped with rapid-fire weapons systems.
The Monte Carlo simulation of this engagement considered the movement,
acquisition, and combat activity (duels) of each and e\/ery element in the
battle.1 Maneuvers, in terms of attack speed and accelerations, over different
portions of the terrain were considered for each weapon based on preprocessed
terrain analysis. The existence or nonexistence of line-of-sight between
weapons systems for each route to all other weapon systems was used as input.
Preprogrammed target priority tables were used to specify the allocation of
individual weapons to targets. A replication of the simulation consisted of
moving each of the systems down their prespecified paths and evaluating, by
Monte Carlo means, the acquisition and attrition processes (the fundamental
duel event) for each weapon system during the course of the engagement. The
engagement was replicated many times to obtain a level of statistical stability
for the results.
The hybrid combat model was applied to this and other engagements by
aggregating individual weapons systems into groups. Thus, for each route on an
axis there were two separate groups of main battle tanks or armored personnel
carriers (APCs)„ The long-range missiles were aggregated into one group and the
short-range missiles were aggregated into three groups, one for each axis.
The Red defensive force was aggregated by weapon type for each axis, thus
producing nine Red defensive groups. Also included, but not shown in the
figure, were indirect-fire artillery weapons systems for both forces.
Using the attrition-rate models discussed in section 3.1, the attrition
rates for each group on appropriate target groups were calculated using the
same basic firing time, accuracy, and lethality data used in the simulation.
Target acquisitions were determined with the parallel acquisition model using
the same detection rate and pinpoint probabilities used in the simulation. The
allocation factors (e.. and h..) employed were based on the priority tables
used in the simulation. The intelligence factor was set equal to 1.0 since
these effects were not considered in the simulation. Mobility and line-of-
sight data from the preprocessor were considered in a deterministic manner
similar to that employed in the simulation. Average speeds and line-of-sights
over segments of the routes were input for each of the aggregated groups.
Some of the engagements considered as many as 100 individual weapon systems.

40
SETH BONDER
Thus, a group was moved as a whole, and visibility did or did not exist to the
group as an entity. The differential equations were solved numerically using
computational time-step procedures.
Using this approach, the model was applied initially to short-range
defense and long-range attack scenarios considered in a specific study
program. With these engagement types, runs involving different weapon systems
and force structures were made for comparison with the simulation results. Some
of these comparisons are shown below in tables 1 and 2.
Table 1 presents a comparison of the results of one of the short-range
defense engagements. The initial numbers of forces and the numbers of survivors
at three analysis points as predicted by both Monte Carlo simulation and the
hybrid model are given. The analysis points are defined by the percentage of
Red tank survivors: low equal to 70 percent, principal equal to 50 percent,
and high approximately equal to 20 percent. The times at which these analysis
points are reached in each of the models also is given. Two sets of results at
the low analysis point in the hybrid model are shown since there was an
appreciable attrition in the 240-250 time interval.
Table 2 presents the comparisons of tank survivors only at the three
analysis points for another short-range defense and one long-range attack
engagement. Table 3 presents comparisons of six pure tank battles in terms of
their loss exchange ratios (LER) at the end of the battle. The LER is the ratio
of enemy (Red) to friendly (Blue) armored system losses and is interpreted as
the expected trading ratio between two specific combatant forces if they were to
engage in many similar battles.
The initial favorable comparisons between this hybrid analytic/simulation
of battalion level combat activities and the detailed Monte Carlo simulation
strongly supported the hypothesis that both models were essentially describing
the same combat process. This, and hundreds of additional comparisons since
then, have given credence to this model and its many successive versions developed
over the past ten years. These models have been used extensively during this
period, providing information to address many system requirements and system
choice issues. Additionally, the models have provided some useful insights
regarding tactics for the defense of Western Europe.
Current estimates are that the Warsaw Pact force could attack the NATO
Alliance along the West German border with armored force ratios ranging from
3:1 to 6:1. Analysis of many small unit actions indicates that in a single
battalion- or company-sized defensive engagement, the instantaneous loss
exchayige ratio1 as a function of battle time is as depicted in figure 6.
Mathematically, the ratio of the rates of attacker and defender losses.

MATHEMATICAL MODELING OF MILITARY CONFLICT SITUATIONS
41
TABLE 1
COMPARISON OF SURVIVING FORCES
Short-Range Defense
Initial Numbers
16 Blue Tanks
6 Blue Short-Range Missiles
6 Blue APC
3 Blue Long-Range Missiles
40 Red Tanks
0 Red Missiles
12 Red APC
ANALYSIS
POINT
Low
(70%)

Principal
(50%)
High
(22%)
WEAPON
Blue Tanks
Blue SR Missi
Blue APC
Blue LR Missi
Red Tanks
Red Missiles
Red APC
Blue Tanks
Blue SR Missi
Blue APC
Blue LR Missi
Red Tanks
Red Missiles
Red APC
Blue Tanks
Blue SR Missi
Blue APC
Blue LR Missi
Red Tanks
Red Missiles
Red APC
les
les
les
les
les
les
SIMULATION
13.90
5.10
5.93
2.73
28.00
11.70
12.23
4.57
5.73
2.27
20.00
10.33
9.40
2.97
5.20
2.00
8.90
4.27
TIME
242
263
327
HYBRID
ANALYTIC/
SIMULATION
15.1/13.9
6.0"
6.0"
3.0'
30.4/24.4
11.0/10.6
12.6
6.0'
6.0"
3.0"
19.2
10.2
10.0
5.8
6.0"
2.9
7.2
7.0
TIME
240/250
260
290

TABLE 2 COMPARISON OF SURVIVING TANKS
BLUE FORCE MISSION
Short-Range Defense
Long-Range Attack
WEAPON
(Time)
Blue Tank
Red Tank
(Time)
Blue Tank
Red Tank
INITIAL NO.
19
40
31
13
LAP
SIM.
(242)
17.20
28.00
(206)
26.30
9.00
HYBRID
(240)
18.0
30.4
(260)
27.6
9.1
PAP
SIM.
(259)
15.87
20.00
(411)
23.83
7.0
HYBRID
(260)
15.4
17.6
(440)
23.8
7.0
HAP
SIM.
(352?)
13.33
8.0
(512)
21.47
2.0
HYBRID
(280)
13.7
8.8
(470)
23.5
1.4

MATHEMATICAL MODELING OF MILITARY CONFLICT SITUATIONS 43
TABLE 3 COMPARISON OF LOSS EXCHANGE RATIO
FOR PURE TANK BATTLEb
Short
Range
Medium
Range
Long
Range
Short
Range
Medium
Range
Long
Range
\
T
A
C
K
E
F
EN
s
E
1 *"
Blue
54
54
54
46
48
38
■ ' ' ' ■ ' ■
itial Number of
Tanks
Red
40
40
27
72
72
72
Ratio
Red/
Blue
.74
.74
.50
1.57
1.50
1.89
| — 1 . - M 1
Loss Exchange Ratio
i at End of Battle
1
, —iLJ— -——._
Simulation
1.08
1.10
.72
2.10
1.95
2.00
Hybrid
Analytic/
Simulation
.92
1.10
.96
2.10
1.68
1.87

^
^
INSTANTANEOUS
EXCHANGE
RATIO
CHANGE IN ATTACKER LOSSES
CHANGE IN DEFENDER LOSSES
CO
m
CD
o
m
TO
INITIAL FORCE RATIO
(ATTACKERS/DEFENDERS)
BATTLE TIME
FIGURE 6 INSTANTANEOUS EXCHANGE RATIO AS A FUNCTION OF BATTLE TIME

MATHEMATICAL MODELING OF MILITARY CONFLICT SITUATIONS
45
The instantaneous exchange ratio is very high and relatively independent of the
force ratio (and particularly of threat size) early in the battle because of
concealment and first shot advantages accrued the defender. The instantaneous
exchange ratio advantage moves to the attacker as the forces become decisively
engaged, because more attackers find and engage targets and the concentration
and saturation phenomena come into play for the attacker.
In terrains that permit it, this suggests that an in-depth use of a
large number of company and below, brief, direct fire engagements or ambushes
may be an effective tactic. Such a campaign might occur in two phases.
Phase I: This phase involves a somewhat constant density of
small mechanized antitank (AT) teams who engage threat
forces in carefully selected (and perhaps prepared)
terrain locations. They operate at the very high,
essentially force ratio independent, part of the
instantaneous exchange ratio curves by firing a
small number of rounds and immediately withdrawing
before suffering any attrition. The withdrawal is
made to another engagement site, passing through a
similar site occupied by another AT team.
Phase II: This phase continues the sequence of engagements of
phase I; however, increasing amounts of enemy
attrition are to be obtained per unit of ground traded at
the expense of defender losses. This is achieved by
increasing the density of AT teams or by organizing
increasingly larger combat units who engage for longer
periods of time (thus moving farther down the
instantaneous exchange ratio curve). This sequence of
progressively hardening the defense continues until
the final boundary line at which time units are
organized for the "decisive battle" if necessary.
Rather than focus on obtaining good force ratios for individual engagements
as do other tactical concepts, this tactical concept attempts to make
use of conjectured, early-on, high, instantaneous exchange ratios. A number
of weapon system acquisitions and tactical changes in the European theater
are based on these concepts.
4.0 OVERVIEW OF CURRENT STATUS AND DEVELOPMENT TRENDS
As previously noted, utility of the initial differential hybrid analytic/
simulation model of small unit engagements gave rise to a number of variants in

46
SETH BONDER
the late sixties and seventies. The period 1972-1978 produced additional
improvements in modeling combat activities. Using some improved modeling
techniques,1 many of the existing models were improved and new models
developed for the three levels of battles over the past eight years. The
directions of these improvements and associated trends are shown in figure 7.
Although high resolution Monte Carlo simulations are still used at the
battalion level, a large number of hybrid analytic/simulation models have
been developed and are extensively used in tactical warfare studies of system
design requirements, system choice, and system mix issues. Using the same
kinds of input data, these models have been shown to generate essentially the
same battle results as the simulation models but orders of magnitude more
efficiently. At the division-corps level, a number of the war games have been
improved to reduce player participation, and, more recently, a numter of
hybrid analytic/simulation models have been developed. At the theater level,
simplified "firepower score" analytic structures have been replaced by first
simple and then more complex hybrid analytic/simulations. Finally, over the
last year or two, some initial analytic techniques for aggregating and
extrapolating results of the more dynamic hybrid analytic/simulation models
and war games at the corps and theater level have been developed.
It is interesting to note the trends that appear to have occurred at
each of the levels of tactical warfare modeling. At the battalion level the
models have progressed from simple Monte Carlo and analytic duels to very
detailed simulatory structures and appear to be moving more in the analytic
direction with complex hybrid analytic/simulations. At the division-corps
level the progression has been from simplified manual and computer-assisted
war games to high-resolution detailed games, and then to detailed hybrid
analytic-simulations. At the theater level the progression has been from
simplified war games, to simplified analytic models, to much more complex
high resolution hybrid analytic/simulations. Considering all of the
developments over the twenty-some odd years, the trend is similar to that observed in
a number of other modeling areas: initial developments are rather simplified,
mostly analytic models which progress to \jery detailed high resolution structures
that are somewhat simulatory in nature, and then more sophisticated analytic
structures are used to describe the complexity and detail. Accordingly it is
my impression that in the next decade the developments in each of the three
areas will move in the direction shown on figure 7. I believe that an
essentially pure analytic model of battalion level activities will be developed,
xThe interested reader is referred to Bonder (1978) for an overview description
of advances in modeling technology which occurred in this period.

BATTALION & BELOW
MONTE CARLO/ SIMPLIFIED
ANALYTIC DUELS MONTE CARLO
SIMULATIONS
DIVISION-CORPS LEVEL
HIGH RESOLUTION
MONTE CARLO
SIMULATIONS
HYBRID
*~ANALYTIC""
SIMULATIONS
ANALYTIC
3
o
3>
O
o
m
MANUAL WAR
GAMES
THEATER LEVEL
SIMPLIFIED
COMPUTER ~
ASSISTED
WAR GAMES
SIMPLIFIED (FIREPOWER SCORE)
& OTHER WAR GAMES
COMPLEX COMPUTER
"ASSISTED WAR GAMES
REDUCED PLAYER
- PARTICIPATION -
WAR GAMES
SIMPLE S COMPLEX
HYBRID ANALYTIC/SIMULATIONS .
I
ANALYTIC AGGREGATION *
EXTRAPOLATION TECHNIQUES
FIREPOWER SCORE & OTHER
SIMPLIFIED ANALYTIC MODELS"
SIMPLE HYBRID
' ANALYTIC/SIMULATIONS
COMPLEX HYBRID
-ANALYTIC/SIMULATIONS *
en
o
INTERACTIVE MODELS
MORE ANALYTIC HYBRID
MODELS
aggregation/extrapolation
techniques
O
O
1956
1960
1962
1964
1966
1968
1970
1972
1971
1976
1978.
o
GO
FIGURE 7 TRENDS IN TACTICAL WARFARE MODEL DEVELOPMENTS
o
z
CO
-3

48
SETH BONDER
possibly by 1985. The larger-scale battle models at the corps and theater
level will probably move in three related directions. The hybrid analytic/
simulation models will become even more analytic in structure. Because of the
difficulty in modeling the command behavior of multiple players, player
participation in war games will be further reduced until the games are in essence
interactive models. Finally, the analytic aggregation/extrapolation techniques
will be further improved to utilize the results of the hybrid and interactive
models.

MATHEMATICAL MODELING OF MILITARY CONFLICT SITUATIONS
49
5.0 BIBLIOGRAPHY
Adams, H.E., et al., "Carmonette: A Computer-Played Combat Simulation,"
Technical Memorandum 0R0-T-389, Operations Research Office, The Johns
Hopkins University, February 1961.
Ancker, C.J., and Gafarian, A.V., "The Distribution of Rounds Fired in
Stochastic Duels," Naval Research Logistics Proqramminq 11, 4 (1964)
303-327.
Ancker, C.J., Jr., and Williams, T., "Some Discrete Processes in the Theory of
Stochastic Duels," Operations Research 13, 2 (1965) 202-216.
Ancker, C.J., Jr., and Gafarian, A.V., "The Distribution of the Time Duration
of Stochastic Duels," Naval Research Loqistics Quarterly 12, 3 and 4
(1965) 275-294.
Ancker, C.J., Jr., "Stochastic Duels of Limited Time Duration," Canadian
Operational Research Society Journal 4, 2 (1966a) 59-81.
Ancker, C.J., Jr., "The Status of the Development of the Theory of Stochastic
Duels-II," SP-1017/008/01, System Development Corporation, Santa Monica,
CA, September (1966b).
Bach, R.E., Dolansky, L., and Stubbs, H.L., "Some Recent Contributions to the
Lanchester Theory of Combat," Operations Research 10, 3 (1962) 314-326.
Barfoot, C.B., "The Lanchester Attrition Rate Coefficient: Some Comments on
Seth Bonder's Paper and a Suqgested Alternate Method," Operations Research
17, 5 (1969) 888-894.
Benjamin, W.C, and Gholston, W., Compilation of Results Obtained from Some
U.S. Firings of Kinetic Energy Projectiles aqainst Tanks (U),
Confidential, Ballistics Research Laboratories Memorandum Report No. 1295,
August 1960.
Bonder, S., "The Lanchester Attrition-Rate Coefficient," Operations Research
15, 2 (1967) 221-232.
Bonder, S., "Mathematical Models of Combat," in Topics in Military Operations
Research, The University of Michigan Engineering Summer Conferences, 21
July-1 Auqust 1969.
Bonder, S., "The Mean Lanchester Attrition Rate," Operations Research 18, 1
(1970) 14-23.
Bonder, S., "Perspectives from Defense Modeling," keynote address presented at
Washington Operations Research Council Symposium, Models in Government:
New Developments and Applications, Washington, DC, 20 October 1978.
Bonder, S., and Farrell, R.L., (Eds.), Development of Models for Defense
Systems Planning, Report No. SRL 2147, TR 70-2, Systems Research
Laboratory, Department of Industrial Enqineerinq, The University of
Michigan, September 1970 (National Technical Information Service,
Springfield, Virginia, Report No. AD 715 664).
Bowen, K.C., "Analytical Models for the Study of Battle Outcome Distributions,"
Third Tripartite Naval Operational Research Symposium, March 1963.
Brackney, H., "The Dynamics of Military Combat," Operations Research 7, 1
(195D 30-44.
Brown, R.A., "A Validation Study of Certain Combat Models," Unpublished Working
Paper, Systems Research Group, The Ohio State University, Columbus, Ohio,
(1959).
Brown, R.H., A Stochastic Analysis of Lanchester's Theory of Combat, ORO-T-323
(U), Operations Research Office, The Johns Hopkins University, Chevy
Chase, MD (1955).
Brown, R.H., "Theory of Combat: The Probability of Winning," Operations
Research 11 (1963) 418-425.

50
SETH BONDER
Clark, G., "The Combat Analysis Model," Ph.D. Dissertation, Department of
Industrial Engineering, The Ohio State University, (1968).
Clarke, B.C., "The Offensive Employment of Tanks," Armor LXXI 3 (1962) 42-43.
Combat Developments Command, Armor Agency, U.S. Army, "Tank, Antitank, and
Assault Weapons Requirements Study," Fort Knox, KY: USACDC Armor Agency
(1969).
Cramer, H., Mathematical Methods of Statistics, Princeton University Press,
Princeton, MJ (1961).
Deitchman, S.J., "A Lanchester Model of Guerrilla Warfare," Technical Note
62-58, Institute for Defense Analysis, October 1962.
Dolansky, L., "Present State of the Lanchester Theory of Combat," Operations
Research 12 (1964) 344-358.
Dresher, Melvin, Games of Strategy - Theory and Application, Prentice-Hall,
Englewood Cliffs, NJ (1961).
Engel, J.H., "A Verification of Lanchester's Law," Operations Research 2
(1954) 163-171.
Evans, G.W. II, Wallace, G.F., and Sutherland, G.L., Simulation Using Digital
Computers, Prentice-Hall, Inc., Englewood Cliffs, NJ (1967), Chapter 4.
Freedman, R., "Vulnerability Procedures for Direct-Fire Projectiles," Chapter
3 in The Tank Weapon System, Report No. RF 573 AR 65-1 (S) Systems
Research Group, The Ohio State University, Columbus, Ohio (1965).
Feller, W., An Introduction to Probability Theory and Its Application,
John Wiley and Sons, Inc., New York, NY (1957).
Gause, A., "Command Techniques Employed by Field Marshall Rommel in Africa,"
Armor LXVII 4 (1958) 22-25.
Gnedenko, B.V., The Theory of Probability, Chelsea Publishing Company, New
York, NY (1962).
Helmbold, R.L., "Some Observations on the Use of Lanchester's Theory for
Prediction," Operations Research 12 (1964) 778-781.
Helmbold, R.L., "A 'Universal' Attrition Model," Operations Research 14, 4
(1966) 626-635.
Hogg, R. V., and Craig, A.T., Introduction to Mathematical Statistics,
Macmilliam, New York, NY (1959).
Isaacs, R., Differential Games, John Wiley and Sons, Inc., New York, NY (1965).
Kimball, G.F., and Morse, P.M., Methods of Operations Research, John Wiley
and Sons, New York, NY (1951).
Lanchester, F.W., Aircraft in Warfare: The Dawn of the Fourth Arm - Vol. V.,
The Principal of Concentration, Engineering 98 (1914) 422-423.
Marshall, C.W., "Probabilistic Models in the Theory of Combat," Transactions
of the New York Academy of Sciences, Ser. II, 27, 5 (1965) 477-487.
Montross, L. and Canzona, N.A., U.S. Marine Operations in Korea, 1950-1953,
II, Washington (1955).
O'Ballance, E., The Sinai Campaign of 1956, Fredrick A. Praeger, New York,
NY (1959).
Parzen, E., Stochastic Processes, Holden Day Inc., San Francisco, CA (1962).
Rapoport, A., "Lewis F. Richardson's Mathematical Theory of War," Conflict
Resolution 1, 3 (1957) 249-299.
Richardson, L.F., Arms and Insecurity, Quadrangle Books, Inc., Chicago, IL
(1960a).

MATHEMATICAL MODELING OF MILITARY CONFLICT SITUATIONS 51
Richardson, L.F., Statistics of Deadly Quarrels, Quadranqle Books, Inc., Chicago,
IL (1960b).
Robison, S.S. and Robison, Mary L., A History of Naval Tactics from 1530 to
1930, U.S. Naval Institute, Annapolis, MD (1942).
Saaty, T.L., Mathematical Methods of Operations Research, McGraw-Hill Book
Co., New York, NY (1959).
Schaffer, M. B., "Lanchester Models of Guerrilla Engagements," Operations
Research 16, 3 (1968) 457-438.
Schoderbek, J.J., "Some Weapon System Survival Probability Models - I. Fixed
Time Between Firings," Operations Research 10, 2 (1962) 155-167.
Snow, R., Contributions to Lanchester's Attrition Theory, Memorandum RA-
15078, RAND Corporation, April 1948.
Taylor, J. and Brown, G., "Canonical Methods in the Solution of Variable-
Coefficient Lanchester-Type Equations of Modern Warfare," Operations
Research 24 (1976), 44-69.
Vector Research, Incorporated, "Modification and Improvement of Differential
Models of Combat," VRI-2 FR 70-1(U), Vols. I and II, Vector Research,
Incorporated, Ann Arbor, Michigan, October 1970.
Wagner, H., Principles of Operations Research: With Applications to Managerial
Decisions, Prentice-Hall, Inc., Englewood Cliffs, NJ (1969).
Weiss, H. K., "Lanchester-Type Models of Warfare," Proceedings of the First
International Conference on Operational Research," Operations Research
Society of America, Baltimore, MD (1957) 82-98.
Weiss, H.K., "The Fiske Model of Warfare," Operations Research 10 (1962)
569-571.
Weiss, H.K., "Review of Lanchester Models of Warfare," Paper presented at
30th National Meeting of the Operations Research Society of America,
Dunham, North Carolina, October 1966.
Whitney, D.R., Elements of Mathematical Statistics, Henry Holt and Co.,
New York, NY (1959).
Willard, D., Lanchester as Force in History: An Analysis of Land Battles of
the Years 1618-1905, RAC-TP-74, Research Analysis Corporation, November
1962.
Williams, T., "Stochastic Duels - II," SP-1017/003/00, Systems Development
Corporation, Santa Monica, CA, September 1963.
Williams, T. and Ancker, C.J., "Stochastic Duels," Operations Research 11,
5 (1963) 803-817.
Zeller, G.A., Methods of Analysis of Terminal Effects of Projectiles against
Tanks (U), Secret, Ballistics Research Laboratories Memorandum Report No.
1342, April 1961.
VECTOR RESEARCH, INCORPORATED
P0 BOX 1506
ANN ARBOR, MICHIGAN 48106

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Proceedings of Symposia in Applied Mathematics
Volume 25, 1981
QUEUEING NETWORKS
Ralph L. Disney
ABSTRACT. The study of queueing networks is a rather recent
addition to queueing theory. In this paper we will review
about the last 20 years of developments in the area. Primary
emphasis is placed on Jackson networks and the major
contribution of Kelly. We discuss queue length processes, waiting
time processes, busy period processes, and departure processes.
We will also note a few new studies that have appeared in the
area of network flows. Topics that are not discussed in detail
in the paper are briefly noted in the final section. The
bibliography can serve as an introduction for further reading
in the area.
1. Introduction and Some Background
1.0 Introduction. In this paper we will briefly review some developments
that have occurred over the last 20 years, in an area of applied mathematics
called queueing network theory. The need for such applied work has been
clear almost since the beginning of what is called queueing theory. Some of
the work of A. K. Erlang (see Brockmeyer, et al. [1948]) seems to be pointing
toward a study of interconnected systems of service centers. Certainly by
the 1930Ts it was clear that such work was evolving. (See chapters 7-10 in
Syski [I960].) In most of these early studies, the developments appear to
be closely tied to attempts to solve problems that occur principally in
telephone switching systems.
1980 Mathematics Subject Classification 60 K25, 90B22.
This research was supported jointly by NSF Grant ENG77-22757 and by the
Office of Naval Research Contract N00014-77-C-0743 (NR042-296). Distribution
of this document is unlimited. Reproduction in whole or in part is permitted
for any purpose of the United States Government.
Copyright © 1981 American Mathematical Society
53

54 RALPH L. DISNEY
Much of classical queueing theory (up to about 195 7 or 1960) was
concerned with properties of Markov processes and except for names given to the
parameters and processes it is difficult to separate out the peculiarities of
queues from other Markov processes that were being used as models, for example,
in biological modelling. In fact, many of the birth-death models of biological
studies have interesting and useful counterparts in queueing theory. For this
reason a carefully drawn history of queueing theory would be strained because
of the interrelations with other fields. The useful thing to remember is that
in its beginning phases the field was closely allied with attempts to solve
real life problems.
The 1950*s were perhaps the "golden age" of developments in the field.
Under many impacts (including the introduction of more sophisticated and
realistic models into problems that heretofore had been the domain of the
industrial engineer) the field began to take on a life of its own. In much of the
1950-60 work and extending even to today, the work in the field veered more
and more away from its applied bent. It was at this time that queueing theory
started to be sufficiently arcane that many researchers were turned away from
it. Easy queueing problems had been solved. Many papers were published on
special cases of birth-death processes. The field was moribund after about
1965-70.
Starting about 1965, the field of computer systems analysis began to delve
into properties of time sharing and interconnected systems. Here the
researchers encountered many of the same types of problems encountered
earlier in telephone systems and production systems. It was natural,
therefore, to use existing, known results. By 1970 and certainly by 1973 research
into computers and computer systems had uncovered a large number of problems
beyond the work in earlier queueing theory.
At the same time that these developments in computer systems were taking
place, new problems were occurring and old unsolved problems were becoming
more pressing in telecommunication theory and production theory. New areas
of application were evolving in military command and control modelling;

QUEUEING NETWORKS 55
disease modelling, military tactics modelling, public sector models for
police, fire, and medical emergency systems design; and many others. All of
these topics put demands on queueing theory (and indeed other areas of
probability and random processes) which that theory was unable to handle.
Systems (e.g., networks) of queues rather than single queues became areas
of important study in many of the above applications. Markov process theory
did not provide a rich enough body of knowledge so new ideas evolved.
Largely because the applied topics put demands on existing theory that
could not be accommodated quickly enough, much of the applied work turned to
Monte Carlo simulation of systems. But that placed a new load on the theory
of statistics as well as probability, random processes, and queueing. Because
of the relative quiescence of the 1957-70 period, queueing theory lost ground
to applied topics.
1.1 Purpose. The purpose of this paper is to briefly summarize where we stand
in the study of queueing network theory. Time and space do not permit any in-
depth discussion here. The bibliography should be consulted by the reader
interested in "reading themselves into" the area. However, that reader should be
forewarned that papers on these topics are spread over a large number of
journals throughout the world. There is no one best place to look for research
papers or to expect to follow diligently and thereby stay up with the field.
For background into queueing theory the reader might consult Syski [1960] who
does a masterful job summarizing work in queueing theory up to about 1960.
KleinrockTs two volumes [19 75; 19 76] summarize many known results in queueing
theory and give a nice discussion of many of the problems of queueing phenomena
occurring in computer systems. One can find some discussions concerning
section 2, to follow, in most post 1975 texts. Material in section 4 is
nicely presented in Kelly [1979], Topics in section 5 and 6 have not been
pulled together in any one place to the best of our knowledge so the reader
is on his own there. References are provided herein but that list is far
from complete.

56
RALPH L. DISNEY
1.2 Some Background, Notation and Symbolism. In order to embed our future
discussion in the more classic field of queueing theory, we will present a
brief review of a few ideas in basic queueing theory.
At an intuitive level queueing theory is concerned with problems arising
in waiting lines. One supposes that there is something that can perform a
needed service, a server. In the simpler cases it is assumed that the sequence of
service times is a sequence of mutually independent random variables that are
identically and non-negatively distributed. In queueing theory such a service
process is called a G-type service process. In the theory of stochastic
processes such a process is called a renewal process.
The demand for this service is usually modelled with the times between
demands being the variables of interest. These times are assumed to be a sequence
of independent, identically distributed random variables that are non-negative
(another renewal process). Such an arrival process is called a Gl-type arrival
process. It is usual to assume that the arrival process and service time
process are independent processes. Such a queueing system is called a GI/G/l
queue. The last argument specifies the number of servers in the system.
Queueing or waiting occurs in such systems whenever an arrival occurs to
find the server already engaged serving a previous arrival. If an arrival
occurs and the server is not so engaged, that arrival immediately enters service,
under the usual assumptions, (There are studies that do not include this
assumption. They are called queueing with set up.)
When service is completed, the next unit in the queue to be serviced is
chosen and under the usual assumptions, that unit immediately goes into service.
The rule used to choose the next unit to serve from among all those in the
queue is called the queue discipline. The most common assumption in queueing
theory is that the first unit in the waiting line is the first customer to be
served. Such a discipline is called a first in-first out discipline. For other
applications other disciplines have been considered. (In military studies the
next unit to be served may be the one posing the most immediate threat. In a
computer center, jobs may be assigned to classes of importance - called priority

QUEUEING NETWORKS 57
classes - and the next job chosen is from among those with the highest priority.
In production systems the next job to manufacture may be the one whose promised
delivery date - the so called "due date" - is closest.)
There are four basic processes created by the interaction of service times,
interarrival times and queue discipline that are of particular importance though
others have been studied. The length of the waiting line, called the queue
length process, {N(t): t _> 0}, is one of the basic processes of concern. The
length of time from entry to exit of the queueing system, {Q : n=l,2,***},
called the sojourn time process, is another of the basic processes. The time from
first entrance of a unit to an idle server until first entrance of the next
unit to an idle server, {B : n=l,2,,,#}, is called the busy cycle. The time
between consecutive exits from the server, {X : n = l,2,«-«}, is called the
n
departure process.
Somewhat more formally we can define the problems as follows:
(1) let 0 _< T < T < T • • • be a sequence of real random variables representing
the times at which an arrival occurs to a queueing system. Let A
= T -T ., n=l,2,«««. Assume A = {A } is a sequence of i.i.d. random
n n-1 n
variables (a renewal process). Call A the arrival process to the queue;
(2) let S be a non-negative real random variable. Let S = {S : n=l,2,«»«}
be a sequence of i.i.d. random variables. Call S the service time process
for the queue. We assume A and S are independent processes;
(3) let lfn H(T ) be an indicator random variable taking values 0 (or 1) if
[U, t j n
T does not (or does) occur in [0,t];
n
00
(A) define NA(t) = I lr_ ,(T ). (NA(t): t > 0} is called the arrival
A n=0 I0,tj n A -
counting process.
The four processes of basic concern to queueing theory can be defined
formally in terms of the A and S process. Prabhu [1965] provides a formal
definition of the queue length process* We will not reproduce that here but
rather we assume that everyone has an intuitive understanding of such a process.
The sojourn time process is closely related to the waiting time process for

58
RALPH L. DISNEY
first in-first out disciplines which can be formally stated as
W = W . + S ,-A, if W . + S ,-A >0,
n n-1 n-1 n n-1 n-1 n
= 0 , otherwise.
Then Q = W + S defines the sojourn time of the n arriving customer. The
xn n n J 6
busy cycle process is the sequence of passage times already discussed. If we
define X as the time between the n and (n-1) service completions then the
n
departure process, {X : n=l,2,«««}, can be more formally given as
st
S , if the (n-1) unit leaves someone in the queue,
n
X =
n ■ st
I + S % if the (n-1) unit leaves an empty queue,
n n
Here I is the foreward recurrence time in the arrival process measured from the
n r
s t
time of departure of the (n-1) unit. I is called the idle time of the server.
n —■
1.3 Special Cases. The major thrust of queueing theory for much of its life
has been to compute formulas for the above measures of effectiveness with
special assumptions on the arrival process, service process, queue discipline
or system capacity. For future reference, we will record some results for one
special case here. The following results as well as many of the special cases
are discussed in most standard texts on the subjects. (For example, see Kleinrock
[1975; 1976].)
If we assume that NA(t) is a Poisson process (with parameter A as is usual)
and S is exponentially distributed (parameter u as is usual), the service center
has 1 server, the queue discipline is first in-first out and the system's
capacity is unlimited then the system is called an M/M/l queueing system. A
considerable amount is known about the above four processes (Kleinrock [1975]).
For example:
(1) (N(t)} is a Markov process with state space (0,1,2,•••);
(2) Pr[N(t) = k] is known (but we'll not reproduce the results here.
Essentially the probability is given by an infinite sum of Bessel functions of

QUEUEING NETWORKS 59
type-I, i.e., Bessel functions of the second kind with imaginary argument.);
(3) lim Pr[N(t) - k] - (l-p)pk, k-0,1,2,"' for p - A/u (called the traffic
t-*»
intensity) . Such a limiting distribution exists if and only if p < 1,
otherwise the limit is identically 0 for all finite k. These limiting
probabilities, when they exist, are called the steady state probabilities;
(A) {W } is a random walk on the non-negative reals with a delaying barrier at
0. (See Feller [1963] or Borovkov [19 76].);
(5) Pr[W jc t] is known in principle (through the theory of random walks);
(6) lim Pr[Q < t] = 1-e"^ , t _> 0. This limit exists if and only if
n-*»
p < 1, otherwise the limit is 0 for all finite t _> 0;
(7) the busy cycle process is a sequence of i.i.d. random variables whose
distribution is known (Cox and Smith [1961], for example);
(8) the departure process {X : n =0,1,2,•••} is, in principle, known for all
n from (2);
(9) in the steady state (i.e., t -> °°), when p < 1, {X } is a sequence of i.i.d.
random variables that are exponentially distributed with parameter A.
That is, the departure counting process is a Poisson process with the same
parameter as the arrival process. (See Burke [1956] or Disney, et al.
[1973].)
Results (1), (3), and (9) are useful to our future discussion.
2. Jackson Networks
2.0 Introduction. It was recognized early in queueing theory that a theory of
single server systems was not adequate. Indeed, at least by the 1930fs (and
before if one is not too picky) research in queueing had begun to explore
systems of queues or what we shall call networks of queue. Erlang about 1910 had
studied systems where many identical servers all handled the same arrival
process (the M/M/c model) in telephone systems. A good discussion of these and
multiple server models is given in Syski [1960], especially chapters 5 and 6 as
well as chapters 7 to 10.

60
RALPH L. DISNEY
2.1 Jackson Networks. A general model in this historic mold that has served
almost as the definition of a queueing network was studied in a paper by J. R.
Jackson [1957]. These models and related problems have come to be called
"Jackson networks." His 1957 model assumed:
(1) there are 1 _< M < <» service centers (Jackson did allow servicing center j
to be comprised of more than 1 server. However, to keep the problem simple
we assume throughout that each servicing center has only one server.);
(2) these M servers are arbitrarily connected by arcs over which units travel
instantaneously fast;
(3) to establish how units proceed through the network one defines p,. to be
the probability that a unit completing service at service center i proceeds
M
to service center j for its next service. Then for each i, 1 - I p. . is
j-i 1J
the probability that a unit exiting node i leaves the system. The matrix
P whose elements are p . is called the switching matrix for the network.
Note that these assumptions about switching imply that a unit leaving
service center i chooses the next service center to visit without regard
to any other condition. We call this switching behavior a multinomial
switch;
(4) service times, S ., n=l,2,'*', at center i are i.i.d. random variables.
These times are each exponentially distributed random variables with
M
parameter y.. The sequences {S .} are independent sequences;
i m i=1
(5) there may be many arrival processes but each one is a Poisson process (with
parameter X. if the arrival process is to service center i). These arrival
processes are independent of each other and of the network;
(6) all queue disciplines are first in-first out;
(7) all queue capacities are unlimited.
2.2 The Queue Length Process. The results that Jackson obtained were rather
surprising. Define N(t) to be a vector whose j element, N.(t),
j = l,2,'**,M, is the queue length at the j service center at t.

QUEUEING NETWORKS 61
Jackson's major results were as follows:
(1) (N(t): t _> 0} is a vector-valued Markov process;
(2) lim Pr[N (t) = k , N2(t) = k2'--N (t) = k ]
t-KO
= lim Pr[N (t) - k ]Pr[N2(t) = k2]••-Pr[N^(t) - k ].
t-*»
That is, the queue length processes in these Jackson networks are
asymptotically (t-x»), mutually independent;
k.
(3) lim Pr[N.(t) - k.] - (l-b.)b. J, j = 1,2, •• • if and only if b . < 1, other-
t-x» J J J J J
wise this limit is 0. Here b. = a./y. and a. satisfies the so-called
J J J J
traffic equation
a = X + Pa.
a is the M-vector of a., X is the M-vector of X , and P is the switching
matrix. We assume throughout that the traffic equation has a positive
solution (or alternatively the maximum eigenvalue of P is less than 1).
In elemental form, this equation says simply that a. is the rate of
arrivals to service system j. This arrival rate is simply the sum of
exogeneous arrivals to system j, (X.), plus the sum of all arrivals to j
from inside of the system, (Pa)..
It is instructive to spend a moment looking at what Jackson said and did
not say. Many papers have been written on Jackson networks that misinterpret
and misuse his results. The major confusion occurs around the third result.
Remember that in the example of section 1.3 we showed that the queue length
process was asymptotically (t-*») geometrically distributed with a parameter
p = X/y < 1. If p _> 1 the limit is zero. Also recall that X was the parameter
associated with the Poisson arrival process and y was the parameter associated
with the exponential service times.
Now result (3) above is also asymptotically geometrically distributed with
a parameter a. that is acting as an arrival rate and a parameter y. acting (is)
as a servicing rate. Ihis result led Jackson to state "This theorem (our
results (2) and (3) above) says, in essence, that at least so far as steady

62
RALFH L. DISNEY
states are concerned (t-*»), this system with which we are concerned behaves
^s_ i_f its departments (our centers) were independent, elementary (i.e., M/M/l)
systems •••". So far, so good. The £s_ ^f emphasis is Jackson's and as it
stands is innocuous enough. Jackson makes one more statement about his results
that may have caused an enormous amount of confusion. He says "This conclusion
is far from surprising in view of recent papers by E. J. Burke (that should be
P. J. Burke) and E. Reich." It appears from his references that Jackson is
here alluding to the result (9) of section 1.3 which is sometimes called
"Burke's Theorem".
Confusion in queueing network literature has occurred by taking the "as
if" of Jackson too seriously. Researchers seem to have taken this to mean that
the queue length processes are independent and are created by M/M/l subsystems.
As a consequence there are papers in the queueing literature that study
queueing networks one node at a time (the independence assumption) as M/M/l queues
(the assumption of Burke's theorem). There are other papers that study sojourn
times in networks as sums of independent, exponential random variables. (See
section 1.3, result (6).) Unfortunately, many authors have shown that the flow
of units within the network are, except for special network configurations such
as trees, not Poisson processes and in fact, are not even renewal processes.
Thus each service center in isolation is not only not an M/M/l queueing system,
it is not even a queueing system with a renewal arrival process. That is, for
a time there was serious confusion in the literature as to what ais_ i^f really
meant. The problem is fairly well understood now. Section 5 and its references
discuss the topic more thoroughly.
2.3 Waiting Times. Waiting times in Jackson networks is a rather unexplored
area. Attention has recently turned toward that area but research here has
only started. There are few special cases that have been studied in detail
principally in what are called tandem queues.
We can define a tandem, Jackson network as one having the structure of
section 2.1 with the added features:

QUEUEING NETWORKS
63
(1) there is only one arrival process and it occurs to the first server (In
terms of the section 2.1 scheme X. = X, X = 0, j f 1.);
(2) p = 0 unless j = i + 1 in which case p = 1, j = 1,2,•••,M. (See
figure 1.)
FIGURE 1
A TANDEM JACKSON NETWORK
ARRIVE
>
DEPART
>
SERVICE CENTER
UNLIMITED CAPACITY QUEUE
) DIRECTION OF FLOW
Since the only results, with which we are familiar, are concerned with
first in-first out queue disciplines, we assume this discipline throughout.
We need a bit more symbolism here. So define:
W . = the total time unit n spends waiting in service center i. Call
this the waiting time of unit n in that service center;
Q . = the time spent by unit n in service center i. Call this the
sojourn time of unit n in center i. W . + S . = Q ., i = lt2t--«tM.
—J ni ni ni
n=l,2,---.
The first result is from Reich [1957] which shows: for fixed n, in the
steady state, {Q ,} is a sequence of i.i.d. random variables, each of which
is exponentially distributed. Thus, if Q is the sojourn time in the system
for unit n, this result says simply that this sojourn time, in the limit, is
the sum of independent, random variables
xn xnl ^n2 ^nM
The Q . may depend on y.. Nonetheless the distribution of Q can be obtained
ni l ^n
from simple convolution operations.

64
RALPH L. DISNEY
The next result is surprising. It comes from Burke [1964]. For the
tandem queueing system he shows: the sequence {W .} for n fixed and i = 1,2, •••,*!
ni
is, in the limit, a sequence of dependent random variables. That is, sojourn
times at successive service centers in the network are, in the steady state,
independent but waiting times are not. It is curious that the sequence
{W : i=l,2,'**,M} for fixed n is a sequence of dependent random variables.
But if to each term in this sequence we add a random variable, independent of
{W .} and exponentially distributed (the S .) to obtain a new sequence,
{Q . = W . + S . : i = l,2,'*'}, that sequence is a sequence of independent
random variables for each n.
These waiting time problems exhibit another nasty property that has been
proven in two distinct cases. Burke [1969J proved the first result. To expose
it we must assume that a service center can have more than one server. It is
usual to assume that all service times at a given center are mutually
independent and all are exponentially distributed with parameter u. that depends only
on the center (not on the server in the center - the servers are "identical").
With this set up, Burke considers a three center tandem queue with the first
center having just one server, the second center having any finite number of
servers, (>1), and the third center having just one server. Otherwise, the
problem is that of Reich discussed above. Burke is then able to show that for
each n:
Q and Q „ are independent random variables;
Q n and Q 0 are independent random variables;
nZ iij
Q , and Q . are not independent random variables,
nl n3
These results are surprising. In this network one does not have the Reich mutual
independence. In fact, one does not even have pairwise independence.
Simon and Foley [19 79] have found a similar result to that of Burke in a
different network. In their three service center network, there is one server
at each center. X = X, X. = 0, j =2,3,. The new idea is that the network
is not a tandem queueing network. Rather one has p « = p, p .= 1 - p,

QUEUEING NETWORKS 65
P23 = 1' P3i = °' ^=1»2»3 in the structure of section 2.2. (See figure 2 .)
Then they are able to show that:
Q , and Q « are independent random variables;
nl n2
Q « and Q ~ are independent random variables;
Q 1 and Q ~ are not independent random variables.
FIGURE 2
THE SIMON-FOLEY NETWORK
ARRIVE /^ ^X q . /" "\ DEPART
SERVICE CENTER
UNLIMITED CAPACITY QUEUE
DIRECTION OF FLOW
• SWITCH POINT
One conjectures that what is happening here, in both the Burke problem
and the Simon - Foley problem, is that a unit in queue 2 (in those examples)
may be by-passed by units in server 1 that arrive to the system after unit n.
In both cases, then, the queue length and hence the sojourn time at server 3
will depend on how many such units by-pass unit n, which in turn depends on how
long unit n was in server 1.
Following this line of thought, the following conjecture appears
reasonable. If there is only one path connecting any two single service centers in
the network then one can use the result of Reich [1957] to determine the total
sojourn time of a given unit through a Jackson network. If, on the other hand,
there are multiple paths connecting single server service centers then the
sojourn times of unit n at the service center at the start of these multiple
paths and that at the termination of the multiple paths are dependent.
Furthermore, if there can be multiple servers at any service center then the sojourn

66
RALPH L. DISNEY
times at service centers feeding this multiple server service center and those
being fed by it are dependent unless the multiple server center is first or
last in the sequence.
Unfortunately, we do not know the nature of these dependencies. Neither
do we know the sojourn times through either the Burke or Simon and Foley
networks. This is an area in need of considerably more study.
There is yet one more problem with sojourn times in these Jackson networks.
In perhaps the simplest non-trivial Jackson network one takes all of the
assumptions of section 2.1 with the added assumption of single servers at each service
center. The only alteration is to have just one service station with p.. = p.
(See figure 3.) We call this a queue with instantaneous, Bernoulli feedback.
FIGURE 3
A QUEUE WITH FEEDBACK
INPUT
SERVICE CENTER
| | UNLIMITED CAPACITY QUEUE
> DIRECTION OF FLOW
• SWITCH POINT
OUTPUT
The definition of sojourn time needs a modification. So let: Q - the sojourn
n
time of unit n the i time that unit passes through the server. Then, Q -
the sojourn time of unit n.
12 k
n n n n
where k is a geometrically distributed random variable.

QUEUEING NETWORKS 67
It has been shown by Takacs [1963] and Disney [1978] that for each
fixed n, {Q , i = l,2,,,#}, is a Markov renewal process. Takacs and Disney
each solve the problem for slightly more general cases than Jackson networks
using rather different methods. Once again, as in the Burke or Simon and Foley
examples, the lack of independence seems to be coming from what we loosely
called by-passing.
2.4 Summary of Results. In summary, unlike the Jackson queue length process
results that have been obtained and are elegant in appearance, waiting time and
sojourn time properties are largely unknown. For tandem queues studied by
Reich the problem can be considered solved. For general, single server,
Jackson networks that do not have multiple paths between any two service centers
in the network, it appears that the Reich results can be used to obtain results
easily. For more general networks of Jackson type, the sojourn time problem is
largely unsolved. Because of its importance to many areas it is one of the most
pressing problems in queueing network theory.
3. Busy Periods and Departure Processes
3.0 Introduction. The queue length process and sojourn times have been the
major topics of interest to queueing theory since its beginning. While of some
importance to systems analysis, the busy period and to a lesser extent (until
rather recently) the departure process from single service queues have been of
less importance. The same result is true for Jackson queueing networks as we
shall see below.
3.1 The Busy Period. One can define two related processes of interest. One is
called the busy period. The other is called the busy cycle. The busy period
is the time from entrance of a unit to an empty queue to exit of a unit that
leaves the system empty. The busy cycle is the return time from entrance of a
unit to an empty queue to the next time of entrance of a unit to an empty
queue. If the queue is ergodic such points occur infinitely often. If the
queue is not ergodic such points, except possibly the first one, may not

68 RALPH L. DISNEY
exist. Almost all studies of the busy cycle and busy period with which we are
familiar, assume these times occur infinitely often. (See Cox and Smith
[1961].)
In single server queueing theory in which {A } is a sequence of i.i.d.
random variables, {S } is also a sequence of i.i.d. random variables and {A }
n n
and {S } are independent sequences, the sequence of busy cycles, {B : n=l,2,««.},
is a sequence of i.i.d. random variables if the origin of the time scale is
taken as a point of arrival to an empty queue. The busy period sequence is
not a sequence of i.i.d. random variables. General distributional results are
known for the cases in which the two processes are sequences of i.i.d. random
variables. (For example, see Kleinrock [19 75].)
Unfortunately, the related problems in queueing networks are in a much less
well developed form. To the best of our knowledge there are no known results
for busy periods or busy cycles for Jackson networks. The area could stand
some study.
Stating the problem that needs to be considered is rather easy. Recalling
from section 2.2 (item 1) that {N(t)} is a vector valued Markov process, the
busy cycle process is then simply the time from entrance of an arriving unit
to an empty network to the next time of entrance of an arriving unit to an
empty network. In the busy period case the problem is to determine the time
from entrance of an arriving unit to an empty network to the time at which a
departing unit leaves behind an empty network. We surmise that such points
occur infinitely often for ergodic networks but as pointed out above we know
of no results for the busy period or busy cycle of a Jackson queueing network.
3.2 Departure Processes. There are two problems here that lead to interesting
questions related to our discussion of the queue length process and waiting time
process. One problem is concerned with the nature of the departure processes
from the Jackson network. The other problem is concerned with the nature of
the departure processes from single centers in the network. (Daley [19 76] is
a nice review of these departure processes.) To simplify our discussion we

QUEUEING NETWORKS
69
will assume the network is irreducible in the sense that the switching matrix
P is an irreducible matrix. More general cases have been studied. The
interested reader is referred to Melamed [1979] and its references for these
other cases.
When p. < 1, for i = l,2,#,,,M, every entering unit eventually leaves the
system. Then from Melamed [1979] we have:
(1) in Jackson networks with single server service centers, if j is a node from
which departures from the network occur then the departure process from
the network at this center is a Poisson process with parameter a,;
(2) the collection of Poisson processes of departures from the network are
mutually independent;
The first of these results is quite in keeping with the Burke theorem
(section 1.3, item (9)). The second result is, at first glance, rather
surprising. One would expect that the network itself imposed some dependencies on the
departing processes.
When one turns to consider departure processes from individual service
centers in the network things became a bit more complicated. For the time being
we will continue the discussion within the framework of Jackson networks.
Furthermore, we must distinguish two cases. If p.. > 0 is defined as the n
step transition probability from i to i in the switching process, then there is
some path leading from service center i back to that service center. In this
case we will say that service center i has feedback. Otherwise we will say
service center i does not have feedback. The latter case we can dispose of
quickly.
If service center i does not have feedback, the departure process is a
Poisson process with parameter a..
The service center with feedback requires distinguishing two processes.
(See figure 3 for a picture of the terms used here.) In one process, units
leaving service center i will eventually return to i. In the other process,
units leaving service center i will never return to i. Call the former
process, the feedback stream and the latter the departure stream. (See figure

70 RALPH L. DISNEY
3.) Then again from Melamed [1979] we have:
(1) the departure stream is a Poisson process;
(2) the feedback stream is not a Poisson process and in fact is not a sequence
of i.i.d. random variables.
We can flesh out item (2) a bit more in the case p > 0, that is, feedback
occurs in one step - the so-called instantaneous feedback case. In that case
the output process (figure 3) is a Markov renewal process whose transition
functions are known. Then it has been shown by Disney, et al. [1980] that the
output process is never a Poisson process nor is it a renewal process. However,
the departure process is a Poisson process (parameter X). The feedback process
is not a renewal process. There is reason to believe that the departure
process and the feedback process are not independent random processes but we know
of no proof either way here.
These Melamed and Disney et al. results raise some interesting anomalies
In these Jackson networks, as was noted in section 2.2 (item 3), queue length at
individual service centers act as if they were independent, M/M/l queues. Yet
if the network has feedback loops, the flow on these loops is not a Poisson
process nor even a sequence of i.i.d. random variables. It is this property
(i.e., the distinction between properties of the network and properties of the
individual service centers in the networks) that seems to have created
confusion in some applications of the Jackson network results both in the study of
the queue length process and that of the waiting time process.
A. Extensions to the Jackson Network Theory of 1957
4.0 Introduction. Following his 195 7 paper, Jackson next published a paper in
196 3 in which he followed the basic ideas of the earlier paper. The new ideas
were to allow the arrival processes to the network to be birth processes whose
parameters could depend on the total number of units in the network. Similarly,
the service time processes were death processes with parameters depending on the
number of units at a given service center. In this way the queue length process
(N(t)} becomes a vector valued birth-death process.

QUEUEING NETWORKS
71
4.1 Queue Length Processes in the Vector-Valued Birth-Death Process. We will
not reproduce the exact form of Jackson's 1963 results. They would require
introducing a large amount of new symbolism. Currently available work which
we shall discuss later includes these results. However, it is important to
summarize the findings of Jackson (at a cost of imprecision) because they have
led many others to explorations in queueing networks in an attempt to
generalize the concept of a Jackson network.
In his 1963 paper, Jackson finds the steady state probability vector for
{N(t)}. Much as in the 1957 paper the elements of this vector have a geometric-
like form (called a product form in much of the computing literature). That is,
the elements are of the form
fc1 k? N kM
C bl b2 b3 bM > kj i°'
In the 1957 paper the constant C had the form
(l-b1)(l-b2)...(l-bM)
which leads directly to the results, observations, and comments found in section 2.
In the 1963 case, however, C is not found to be of this form (except in special
cases such as those in the 1957 paper). As a consequence, one cannot infer
that these networks are composed of independent single service centers nor that
these service centers act as if they were simple queues.
Since these results appeared, considerable effort has been expended trying
to determine approximations and easy ways to compute C. Other research effort
has been expended on exploring networks that might have the "product form"
of solution. There are no up to date summaries of the large amount of work.
The Kleinrock [19 75; 1976] books are basic. The papers of Kelly [19 76; 1978]
and Schassberger [1977; 1978] trace some of the work.
4.2 A Generalization. In a 19 76 paper, Kelly significantly generalizes the
concept of Jackson network and provides important extensions to the concept of
"product forms" of solutions. In Kelly [1979], he provides further insights

72
RALPH L. DISNEY
and significantly broader applications of his study. We will follow his 19 76
publication.
Suppose that there can be I types of units entering the network. Units
of type i e I enter the network as a Poisson process with rate v(i) and pass
through the servers according to the path
r(i,l)r(i,2)...r(i,S(i))
before leaving the system. Thus at stage s (s =1,2,••*S(i)) of its route, the
unit is at queue r(i,s).
Within each queue, the units are ordered so that there are units in
positions 1,2,•*,,n.. n. is the total number of units in queue j.
Each unit requires a random amount of service. This service time is an
exponentially distributed random variable with mean 1.
There is a single server who supplies a total service effort at rate
<{>.(n.) in such a way that y.(l,n ) of this effort is directed to the unit in
position £. When this unit leaves the j service system, all units behind it
move up a space.
When a unit arrives at service center j it moves immediately into
location H with probability <5.(£, n.+l). Units formerly occupying spaces £,
I + l,"«,n. are moved to spaces £ + 1, Jl + 2,«",n. +1.
Such a structure is rather general for queueing network behavior. The
queue discipline of the earlier Jackson networks has been considerably
generalized. Arrival processes are still Poisson but note the arrival rate may
depend on the "type" of the unit. "Type" may be associated with the path taken
by the arrival simply by associating an arrival "type" with the path that
arrival will follow. Service times here are also more general.
Another important aspect of the Kelly paper is its method of determining
the limiting probability vector for the queue length random process. Whereas
the earlier Jackson network papers of 1957 and 1963 proceeded from the
structure of the problem to set up the usual limit form of the Kolmogorov
equations of the Markov process {N(t)}, Kelly prefers to work with relations

QUEUEING NETWORKS
73
embodied in a reversed process. It appears that when such an approach can be
made to work, detailed calculations are obviated. The Kelly book [1979]
expands on this point and considers these methods for a wide variety of
problems in multidimensional Markov processes, including the queueing networks
of his 1976 paper.
Kelly starts with a stable, conservative, regular Markov process. (All of
the networks discussed so far have these properties.) Then it is well known
from the theory of such processes that if i, j are vector valued states in these
networks, then a solution to the equations
np = o
with n _> 0 and III = 1 is unique and
n(j) = lim Pr[N(t) = j].
The elements of n are the limiting probabilities - the so-called "steady state"
probabilities - of the Markov process. The elements of P are the transition
rates of the process (Kleinrock [1975]). P is called an infinintesimal
generator.
Now it is known that if {N(t)} is a Markov process in equilibrium (e.g.,
the initial vector for the process is II) there then exists another process -
called the reversed process, {N(-t)}, that is a Markov process in equilibrium.
The reversed process has the same limiting probability vector, n, but its
infinitesimal generator may not be that of {N(t)}. (If the two processes have
the same infinitesimal generator then {N(t)} is said to be reversible.) In
general if q(i,j) is the (i,j) element of the infinitesimal generator of
{N(t)} and qf(i,j) that of the infinitesimal generator of {N(-t)} and if
q(i), qf(i) are the corresponding diagonal elements of the infinitesimal
generators of {N(t)}, and {N(-t)} respectively, then we have
(4.2.1) n(i)q(i,j) = n(j)qf(j,i)
and
q(i) = q'(j).

74 RALPH L. DISNEY
(If the process is reversible these equations are called the equations of
detailed balance.)
The extremely useful result is that for the network set up by Kelly, one
can find q(i,j) and qf(i,j) rather easily. This, along with (4.2.1) and the
uniqueness of E as a probability vector then allows one to determine n.
For his network, Kelly defines a two-tuple c (I) = ((t.(il), s (£)) where
t,(£) denotes the "type" of unit in position I in queue j and s.(it), as
previously defined, represents the position along its path reached by this I
unit in queue j. It is shown that for the vector
Cj = (c (1), c (2)---cj(nj))f
C = (crc2,--.,cM)
is an irreducible Markov process on a countable state space. The infinitesimal
generator for this process and its reversed process are found. Then using the
results on reversed processes stated above, Kelly shows that his network has
the product form of solution which is determined up to a normalizing constant.
4.3 Another Generalization. Kelly generalizes his model one more step and in
the process provides the basis for taking a major step out of the restrictive
assumption of Poisson processes and exponential service times. Unfortunately,
this step has a price to pay.
The first step is to generalize the service time assumption of section
4.2. Now instead of service times being exponentially distributed random
variables, it is assumed that when at queue j, the unit that is at stage s of
its path requires an amount of service that is the sum of Z(j,s) independent,
identically distributed, exponential random variables (rate d(j,s)). That is,
service times are now gamma distributed random variables with parameters Z(j,s)
and d(j,s).
The other assumptions of section 4.2 are retained except that it is
required that
(4.3.1) 6 (£, n^+1) = YjU, Jj+1)«

QUEUEING NETWORKS 75
For this problem the state space of the Markov process of interest must
now become a three-tuple. A state now is defined by t.(£), s.(£), as in section
4.2, and x.(it) which denotes the phase of service currently occupied by the
unit. Then, on this three-tuple space one can define a Markov process whose
states are the three-tuples. This process is irreducible and the state space
is countable. The process has a reversed process; its infinitesimal generator
and that of the reversed process can be found. As before the properties of
reversing are used and it is shown that the process has a product form of
solution.
4.4 A Major Generalization. Were the Kelly results to stop here they would
make interesting, perhaps useful, contributions to the theory of Jackson queue-
ing networks. The restriction of Poisson arrival processes and either
exponentially distributed or gamma distributed service times would preclude a wide
spread use of the results (although much of the computer systems analysis
literature finds these conditions to be reasonable for many computer studies).
But more is available.
Under the conditions of the model of section 4.3, Kelly conjectures that
his results can be extended to include G-type service times. (See section 1,2.)
The conjecture is based on a result of Whitt [1974] which shows that finite
mixtures of gamma distributions are dense in the set of arbitrary non-negative
distributions. Though Kelly does not prove that his model of section 4.3
extends to G-type servers, it is proven with the requisite care by Barbour
[1976].
Finally, Kelly drops the Poisson arrival process assumption that has run
through his work. Commenting on his models (described in our section 4.2 and
4.3) he notes that one can allow these arrival processes to be birth processes
whose parameter depends on the total number of units in the system. Recall that
this step was made in the Jackson [1963] paper.
4.5 Comments. The Kelly work probably represents the state-of-the-art in the
study of queueing networks originally arising out of the papers of Jackson.

76
RALPH L. DISNEY
Work continues on these problems and the Kelly work will probably be mined for
quite a while. If one could drop condition (4.3.1) from the Kelly model and
still be able to compute an equilibrium solution, one would have a major
contribution to the theory of queueing networks.
Concerning waiting time, busy period, and busy cycle analysis we know of
no results presently available. Concerning departure processes, Kelly presents
some results on the departures from the network. These processes are
independent, Poisson processes for the models studied in our sections 4.2 and 4.3.
It has been noted by many authors (for example, see Kelly [19 76]) that
since the Jackson limiting probability vector depends on the assumptions of the
arrival process and service time process only through the expected values of
these processes, similar results may well hold for more general arrival and
service time processes. That is, results such as those obtained may be
independent of the distributional assumptions of the model. Schassberger
[1978] explores this topic in more detail and provides a bibliography for
further reading. He calls this independence property, insensitivity.
5. Flow Processes
5.0 Introduction. As soon as one moves very far from the Jackson assumptions
given in section 2, analysis of queueing networks become difficult, at best. Yet
in many applications these assumptions are inappropriate. One would like to
extend the Jackson model.
Relaxing the assumptions presents enormous difficulties. In many cases,
of course, one can augment what is meant by a "state" of the process to retrieve
the useful Markov property inherent in the Jackson and Kelly models. But, for
example, if one were to retain all of the Jackson assumptions of 195 7 except
that the service times were allowed to be non-exponentially distributed (and
thereby, the "forgetfulness" property is lost as is the Markov property for
{N(t)}), one requires a 2M-tuple to retrieve the Markov property for {N(t)}.
Furthermore, this 2M-tuple would have the Markov property on a cross product
space. The space of M-tuples representing the queue length at each service

QUEUEING NETWORKS 77
M
center is the non-negative lattice points in E . This space would have a
Cartesian product with the space of M-tuples representing expired service
times at each service center. Elements of these M-vectors take values in
M
the non-negative E space. This cross product space is not an appealing one
to work with even for M = 1.
5.1 Another Approach to Queueing Networks. To extend the basic models
discussed so far it seems that one must adopt a new approach to the entire problem.
One such approach is to decompose the network into smaller subnetworks that,
hopefully, are easier to study. Ultimately, of course, one must recompose
the network.
If one steps back and looks at what the network is doing, it seems as
though these networks can be described basically as a collection of three
operations being performed on a set of arrival processes. We call these
three operations random deletion, random stretching and recomposition. They
are discussed in section 5.2. Queue lengths, busy periods, waiting times are
consequences of these operations and in principle should be approachable if
the consequences of the operations are known.
The basic problem in studying these networks as smaller subnetworks, is
that one does not know the stochastic properties of the arrival process to
any node in the network, except possibly those nodes that serve only as
entrances for exogenous arrivals or for nodes in very special networks (e.g.,
trees). In most cases the operations of the network itself serve to transform
one stochastic process (e.g., the arrival process) into another stochastic
process with rather different properties. For example, the departure process
from an M/G/l/N queue is, except in four cases, never a renewal process. The
arrival process which is a Poisson process, and hence a renewal process, is
transformed by the queueing system into a non-renewal process.
5.2 Operations on Stochastic Point Processes. In an abstracted form one can
view a queueing network as a collection of three operations performed, in
some sequence (defined by the network), on one or more marked point processes

78
RALPH L. DISNEY
(the arrival process). These operations are known in point process theory as:
random deletions, random stretching, and superposition.
To keep the discussion focused, consider the Jackson network. In that
case, the multinomial switches (assumption 3, section 2) essentially operate on
a marked point process (the Poisson arrivals or departures from a given
service center, i) by deleting those points (and their marks) from the point
process that will be arrivals to service center j. The undeleted points
constitute departures from the network or arrivals to some service center, not
j. Thus, the switching matrix in the Jackson network is, essentially, a structure
for randomly deleting points from a marked point process. The related switching
structure in the Kelly network is a more complicated set of rules for effecting
deletions.
Furthermore, in a queueing network one can suppose that the arrival process
to a service center is a marked point process. Then, what the service center
itself does is to randomly delay each point in this process to form a new
marked point process called the departure process. The amount of delay is
the random amount of time that an arrival spends in the service center.
Finally, one can view the arrival process to a service center as a
generalization of what in stochastic point process theory is called
superposition. There are two ways to view the problem. For our purposes one has
several sets of points representing the epochs of arrival to the queue. If
now one forms the union of these sets and then orders these points (increasing
order), the resulting ordered set is called the superposed set. The sets
thus superposed can be called the constituent sets. Superposition is then
concerned with the study of the properties of the superposed set. In nearly
all studies of superposition (there are a few exceptions), the constituent
sets are assumed to be independent marked point processes. The elements within
the constituent sets need not be independent, however.
In the studies such as Jackson [1963], some of the constituent processes
are neither point processes, as that term is usually used, nor are they
independent. Thus, for a more general class of problem, we use the word

QUEUEING NETWORKS 79
recomposition rather than superposition. So the operations we want in the
study of queueing networks are operations that recompose collections of marked
point processes.
Thus, one approach to the study of queueing networks in which {N(t)} the
vector of queue lengths is not a Markov process (e.g., "non-Jackson" networks)
is to look at flows of units in the network as marked point processes and to
look at the network as a collection of the operations of random deletions,
random stretching and recomposition. In this way attention is turned,
temporarily from the queueing properties of queue length, busy periods or
cycles, and waiting times and given to the study of "random flows" in queueing
networks. In this way one can generalize, considerably, the class of operations
given in section 2.0. Ultimately, one must return to the queueing problems
created by these operations.
5.3 Some Relevant Literature. Disney [19 75] is the only overall review, with
which we are familiar, of results concerning this approach to queueing networks.
Of particular interest would be the forthcoming book by Franken, et al. [1980]
concerned with properties of marked point processes in a queueing framework.
The work of Cinlar [1972] is a splendid review of the superposition problem
in a point process setting. The departure process problem is nicely reviewed
in Daly [19 76], We know of no review of the problem of random deletions. The
paper of Daly and Vere Jones [1972] is a good introduction to stochastic point
process theory, random deletions and random stretching.
There are a great many loose ends in these studies of non-Jackson networks.
In most cases much new work has been done on each of the topics covered by the
cited reviews. Little, if any, of the work is as coherent and tidy as the work
on Jackson networks. As a consequence one must search diligently over a broad
spectrum for results on these many related topics.
6. Summary
6.0 Summary. We have attempted to review in a few pages, more than 20 years
worth of research in the field of queueing network theory. To accomplish this

80
RALPH L. DISNEY
in such a short space we have concentrated on two topics: Jackson networks and
flow in networks. Our primary emphasis has been on the Jackson network results.
In areas of application these are the results that are of primary importance.
Under the pressing restrictions of time and space we have concentrated only on
the basic Jackson work and the important Kelly work. While we have alluded to
other work, we have by no means provided a definitive state-of-the-art survey.
Considerable work has been done on Jackson networks in the past 15 years. This
work is to be found principally in the literature of the computer scientist
whose interests in these topics seems to have revitalized that field. We can
only hope that the reader interested in a host of results and fascinating
applications will consult this literature. The best starting point is probably
the two volumes of Kleinrock [1975; 19 76] and especially the interesting
chapters 4, 5, 6 of volume II which present some of the basic queueing problems
occurring in computer networks as well as an interesting discussion of trials
and tribulations of applying known results to the design of a large scale
system. Beyond that we can only suggest that the interested reader peruse the
journals of computer science (e.g., _J«jA.^.M. or Acta Informatica) as these topics
continue to be researched and applied.
The study of flow processes in queueing networks is fragmented at present.
There are many results. We have mentioned a few. There is much that we have
not said and much that remains to be said. It is our view that the link up
with the more general field of marked point process theory is natural.
For the study of flow processes in networks, it is natural to think of the
network and its components as operators on random point processes as discussed
in section 5. That view will probably provide greater generality and deeper
insights into these flow processes than is now possible.
There are many other topics in these areas that we have not even mentioned.
The useful computational work of Wallace [1974] and Neuts (for example, Neuts
[1979]) has been left untouched. The study of closed networks has been ignored
(e.g., Gordon and Newell [1967]). The interesting network decomposition idea
of Courtois [1978] deserves attention both for its theory as well as its

QUEUEING NETWORKS
81
application potential. The concepts of approximations, including diffusion
approximations and heavy traffic approximation, have not even been mentioned.
One might consult Harrison [1978] to start into this area. It would seem that
there is no end to such an enumeration. The study of queueing networks is an
enormously large and diverse field. Our tutorial has at best "hit the high
spots".
Acknowle dgemen t
I would like to thank Robert D. Foley and Burton Simon for many helpful
discussions in the preparation of this paper.
References
This list of references is intended to be a guide to the literature. If
an author wrote two papers, one a continuation of the other, we only reference
the latter under the knowledge that the first paper is included in the
bibliography of the second. In this way the references can be used to get one started
in the field. More extensive searching would then have to be done by the usual
"follow your nose" principle.
1. Barbour, A. D. (1976), "Networks of Queues and the Methods of Stages,"
Adv. Appl. Prob., 8, 584-591.
2. Borovkov, A. A. (1976), Stochastic Processes in Queueing Theory,
Springer-Verlag, New York,
3. Brockmeyer, E., Halstrom, H. L., and Jensen, A. (1948), "The Life and
Work of A. K. Erlang," Trans. Danish Acad. Tech. Sci., Transactions No. 2.
4. Burke, P. J. (1956), "The Output of a Queueing System," Oper. Res., 4,
699-714.
5. Burke, P. J. (1964), "The Dependence of Delays in Tandem Queues," Ann.
Math. Stat., 35, 874-875.
6. Burke, P. J. (1969), "The Dependence of Sojourn Times in Tandem M/M/s
Queues," Oper. Res., 17, 754-755.
7. (Jinlar, E. (1972), "Superposition of Point Processes," Stochastic
Point Processes: Statistical Analysis, Theory, and Applications (ed. P.A.W.
Lewis), Wiley, New York.
8. Courtois, P. J. (1977), Decomposability: Queueing and Computer Systems
Applications, Academic Press, New York.
9. Cox, D. R. and Smith, W. L. (1961), Queues, Chapman and Hall, London.

82
RALPH L. DISNEY
10. Daley, D. J. (1976), "Queueing Output Processes," Adv. Appl. Prob., 8,
395-415.
11. Daley, D. J. and Vere Jones, D. (1972), "A Summary of the Theory of
Point Processes," Stochastic Point Processes: Statistical Analysis, Theory,
and Applications (ed. P.A.W. Lewis), Wiley, New York.
12. Disney, R. L., Farrell, R. L., and de Morais, P. R. (1973), "A
Characterization of M/G/l/N Queues with Renewal Departures," Mgmt. Sci., 20,
1222-1228.
13. Disney, R. L. (1975), "Random Flow in Queueing Networks: A Review
and Critique," Trans. Amer. Inst. Industr. Engr., 7, 268-288.
14. Disney, R. L., McNickle, D. C, and Simon, B. (1980), "The M/G/l
Queue with Instantaneous Bernoulli Feedback," (to appear, Nav. Res. Log.
Quart., Dec.).
15. Disney, R. L. (1978), "Sojourn Times in Queues with Feedback,"
paper presented at Colloquium on Point Processes and Queueing Theory,
Keszthely, Hungary, Sept. 4-8, 1978. To appear in Proceedings of Keszthely
Conference.
16. Feller, W. (1966), An Introduction to Probability Theory and Its
Applications, vol. 2, Wiley, New York.
17. Franken, P., Konig, D., Arndt, U., and Schmidt, V. (1980), Queues
and Point Processes, Akadamie-Verlag, Berlin (to appear).
18. Gordon, W. J. and Newell, G. F. (1967), "Closed Queueing Systems
with Exponential Servers," Oper. Res., 15, 254-265.
19. Gordon, W. J. and Newell, G. F. (1967), "Cyclic Queueing Systems
with Restricted Length Queues," Oper. Res., 15, 266-278.
20. Harrison, J. M. (1978), "The Diffusion Approximation for Tandem
Queues in Heavy Traffic," -Adv. Appl. Prob. , 10, 886-905.
21. Jackson, J. R. (1957), "Networks of Waiting Lines," Oper. Res., 5,
518-521.
22. Jackson, J. R. (1963), "Jobshop-like Queueing Systems," Mgmt. Sci.,
10, 131-142.
23. Kelly, F. P. (1976), "Networks of Queues," Adv. Appl. Prob., 8,
416-432.
24. Kelly, F. P. (1979), Reversibility and Stochastic Networks, Wiley,
New York.
25. Kleinrock, L. (vol. 1, 1975, vol. 2, 1976), Queueing Systems, Wiley
Interscience, New York.
26. Melamed, B. (1979), "Characterizations of Poisson Traffic Streams in
Jackson Queueing Networks," Adv. Appl. Prob., 11, 422-438.
27. Neuts, M. F. (1978), "Markov Chains with Applications to Queueing
Theory, which have Matrix Geometric Invariant Probability Vector," Adv. Appl.
Prob., 10, 185-212.

QUEUEING NETWORKS
83
28. Prabhu, N. U. (1965), Queues and Inventories, Wiley, New York.
29. Reich, E. (1957), "Waiting Times when Queues are in Tandem," Ann.
Math. Stat., 28, 768-773.
30. Schassberger, R (1978), "The Insensitivity of Stationary Probabilities
in Networks of Queues," Adv. Appl. Prob., 10, 906-912.
31. Simon, B. and Foley, R. D. (1979), "Some Results on Sojourn Times in
Acyclic Jackson Networks," Mgmt. Sci., 25, 1027-1034,
32. Syski, R. (1960), Introduction to Congestion Theory in Telephone
Systems, Oliver and Boyd, Edinburgh.
33. Takacs, L. (1963), "A Single Server Queue with Feedback," Bell Syst.
Tech. J_., 505-519.
34. Wallace, V. L. (19 74), "Algebraic Techniques for Numerical Solution
of Queueing Networks," Math. Me tho ds in Queueing Theory, Lecture Notes in
Economics and Mathematicsl Systems, No. 98, Springer-Verlag, New York.
35. Witt, W. (1974), "The Continuity of Queues," Adv. Appl. Projb. , 6,
175-183.
DEPARTMENT OF INDUSTRIAL ENGINEERING AND OPERATIONS RESEARCH
VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY
BLACKSBURG, VIRGINIA 24061

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Proceedings of Symposia in Applied Mathematics
Volume 25, 1981
PRACTICAL ASPECTS OF FISHERY MANAGEMENT MODELING
1
Frederick C. Johnson
ABSTRACT. The Pacific Coast salmon fisheries are considered to be
the most sophisticated fishery system in the world [1]. In spite of
a great deal of theoretical work [2], [3], [4], the practical
problems associated with the management of these fisheries require a
level of detail far exceeding a general theoretical framework for
optimal resource utilization. While many of the problems are
associated with the characteristic nature of common property resources [5],
there also exist special catch allocation problems arising from
federal court decisions on treaty Indian fishing rights [6]. In this
paper we will first give an overview of the technical and other
problems arising in salmon fisheries management. We will then discuss
the development of a mathematical model for fishing regulation
analysis. This model has become a standard tool used by the Washington
State Department of fisheries to evaluate the economic and biological
impacts of alternative salmon fishery regulation policies.
1. INTRODUCTION
The scientific management of any high seas fishing Industry is faced with
many complex technical, political and economic problems. These technical
problems arise from the basic nature of fishing and fish finding which makes the
determination of fundamental biological parameters such as natural mortality
rates and stock size a difficult and generally imprecise process. The
political and economic problems associated with fishing stem from the fact that
fishing is the sole remaining major food production system which relies on the
hunting and exploitation of wild stocks that are a common property resource.
As a result of the common property nature of a fishing resource, the
conservation, control and allocation of the resource is a highly sensitive political
problem, and agreement among competing users on a unified management policy is
difficult to achieve. The situation is further aggravated by the above
mentioned technical problems which often result in a lack of scientific consensus
1980 Mathematics Subject Classification. 65C20, 68J05, 92A15.
Supported by the Washington State Department of Fisheries.
Copyright © 1981 American Mathematical Society
85

86 FREDERICK C. JOHNSON
on underlying biological parameters, and this lack of consensus provides a
ready made reason to maintain the status quo and to deemphasize important long
term issues.
The common property nature of the fisheries also leads to economic
problems for fishermen due to the phenomenon that good fishing tends to create
more fishermen. The end result can be a seriously overcapitalized fishery in
which only a fraction of the fishing gear is required to harvest the resource
with economic efficiency. Such excess fishing capacity causes severe stress in
fishery regulation for two reasons. Without careful management, the fish
stocks may be severely overharvested, even to the point of extinction. Also,
the practical politics of an overcapitalized situation often makes it more
expedient to restrict the efficiency of the harvestors, by season limitations or
by limiting the types of fishing gear which may be legally used, than to
restrict the number of harvestors. Such limitatations can be highly
controversial.
A final crucial feature of the common property nature of fisheries is that
almost all resource management must be achieved by external regulation rather
than by self-control. The competition for the resource among fishermen means
that it is generally not in the best self-interest of a fisherman to practice
conservation since most of the benefits of his efforts accrue to his
competitors rather than directly to himself.
The Pacific Coast salmon fisheries exhibit all of the problems of a common
property resource. In addition, the State of Washington is faced with special
management problems of mandated catch allocation among certain user groups. A
federal court ruling in 1974 (upheld by the United States Supreme Court in
1979) stated that Indian treaties signed in the 1850's gave the treaty tribes
the right to harvest fifty percent of the salmon originating in Puget Sound and
other areas of Washington State. Since those tribes had been catching only

PRACTICAL ASPECTS OF FISHERY MANAGEMENT MODELING 87
about five percent of these fish, the development of new regulations to meet
the court requirements was a major task. Because of the large number of
alternative regulation sets that seemed feasible, the many different salmon
stocks involved and the complex interrelations between stocks and fisheries,
the development of a mathematical model for salmon fishery management was
selected as the most suitable approach to this regulatory task. The resulting
model has subsequently become a standard management tool in Washington State.
A major goal of the fishery model was to make maximum use of available
data to provide a level of detail adequate to meet management needs.
Fortunately, radically improved fish tagging techniques were available to
provide more information on stock composition of catches than was previously
available. A special challenge was to make good use of these new data.
As will be seen from the subsequent discussion of the model, the basic
equations contained in it are straighforward. The success of the model is
based not on the derivation of complex equations but on the ability to couple
large amounts of data in a systematic analysis which provides useful results
for resource managers. This was achieved by working closely with key
biologists over a period of many months to carefully define the type and quality of
available data, the identify management questions that must be answered by the
model and considering many alternative model configurations. Because of this
lengthy analysis phase and the deep understanding of the biologists, the
subsequent implementation of the model was achieved without major technical
problems.
2. THE SALMON FISHERIES
Salmon are anadromous fish. This means that they reproduce in fresh water
but spend their adult lives in the ocean. Although salmon are highly migratory
and can range over several thousand miles, they have a highly developed homing
instinct, and the vast majority return to their natal streams to spawn. Unlike
the Atlantic salmon all Pacific salmon die after spawning. Because of the

88 FREDERICK C. JOHNSON
strong homing characteristics of the salmon, the concept of a salmon stock is
fundamental to management. A stock of salmon is a subset of a particular
species which has the same stream of origin. Ideally, each stock of salmon is
Individually managed to achieve its escapement goals, that is, to obtain the
required number of spawning fish to reproduce the stock.
This Is a fundamental difference in management from other ocean species.
Because appropriate spawning grounds are limited and fixed, a relatively
accurate determination can be made of escapement requirements. Since there is
only a certain amount of space for egg deposition, overescapement does not
produce more juvenile fish, it results in a waste of the surplus adults. Species
which spawn in the ocean, on the other hand, do not have such a restricted
spawning potential and important questions arise as to the appropriate number
of adults that should be unharvested and left to spawn each year.
Salmon originating in the State of Washington are harvested in sport and
commercial fisheries from Alaska to northern California. Because of the highly
migratory nature of the salmon, regulation changes in one fishing area impact
all of the fisheries throughout the range of the fish. In essence, a
regulation change resulting in an increase (or decrease) of catch in one fishery
results in a decrease (or increase) of catch in the other fisheries in the
migration path of the salmon. Therefore, all of the fisheries must be treated
simultaneously when analyzing proposed regulatory changes.
The management of these fisheries is faced with two other important
factors: hatchery stocks and mixed stock fisheries. Because of the loss of
natural spawning grounds due to logging, construction of dams and urbanization
in general, salmon hatcheries are used as a major source of fish for the sport
and commercial fisheries. A key feature of hatcheries is that natural
mortality, between the taking of eggs and the release of juvenile fish, is
substantially reduced. Consequently far fewer adult fish are needed to meet the

PRACTICAL ASPECTS OF FISHERY MANAGEMENT MODELING 89
spawning requirements of a hatchery stock as compared to a wild stock of equal
size, and the hatchery stock can be much more heavily harvested* In other
words, the harvest rate of a hatchery stock can be much higher than for a wild
stock, and this rate disparity is a major concern wherever wild and hatchery
stocks are simultaneously harvested. Since mixed stock fishing occurs in many
areas, the effects of changes in regulations on escapement needs are vitally
important.
3. THE FISHERY MODEL
The fishery model was developed to serve as a management tool for
analyzing the economic and biological effects associated with changes in salmon
fishery regulations. The goal of the model is to provide a common methodology
for quantifying the impact of alternative sets of fishing regulations on
fisheries performance and salmon stock abundance and stability. The
performance of the salmon fisheries is determined by performing a detailed
calculation of the redistribution among the ocean and inside fisheries in Puget Sound
and the Columbia River of catch, weight and catch value that Is a consequence
of each proposed set of fishery regulations. The evaluation of stock abundance
and stability is based on an escapement analysis which determines whether
specific escapement goals are met on a stock by stock basis. Input data for
the model are used to calculate detailed fishing rates, stock recruitment
populations and other factors such as maturation rates and length/weight
relationships. All of these parameters are then combined with new fishery
regulation data to evaluate the effects of regulation changes.
Regulation evaluation is based on a complete, time-sequenced simulation of
the interactions between the fisheries and the salmon life cycle. This
simulation begins with the initial recruitment of the salmon stocks to the fisheries,
continues through the entire ocean life of each stock, and ends with the final
escapement of the oldest fish. From initial recruitment to final escapement,
the model simulates both the effects of fishing and the biological process of

90
FREDERICK C. JOHNSON
growth, natural mortality, maturation, and migration. The effects of fishing
are based on the population size and distribution of the stocks, their length
characteristics, the regulations controlling the fisheries, and the historical
performance of the fisheries as reflected in catch statistics. The total
catch, weight, and catch value is obtained for each fishery, and the total
escapement is obtained for each stock. This information forms the basis of
comparison between alternative fishery regulations.
The evaluation of fishing regulation changes is ultimately dependent upon
how well alternative regulations satisfy a management policy which consists of
catch distribution and escapement goals. Because of the frequently complex
nature of this evaluation, no attempt was made in the fishery model to compile
an overall measure of worth for each set of regulations. The model simply
computes the effects of regulations—final evaluation remains in the hands of the
management agency.
The fishery model is a steady-state model in which the fishery
regulations, effort, and other characteristics are held constant throughout the life
of the stock. This type of analysis can be viewed as giving the representative
effects of a set of regulations averaged over several years. Such a steady-
state analysis is appropriate for evaluating regulations from a policy
viewpoint. The stochastic components of the stock/fishery system, such as
recruitment failures, bonanza runs, and effort fluctuations, generally cannot be
forecast with sufficient accuracy to be considered from a policy viewpoint. These
variations are more appropriately treated as special cases which (possibly)
require special regulations. In addition, this analysis does not indicate the
effects of the transition period in which a brood year of a stock is fished
under both old and new regulations. The fishery model measures the effects of
a set of regulations after this transition phase has passed.

PRACTICAL ASPECTS OF FISHERY MANAGEMENT MODELING 91
It is important to understand that the objective of the fishery model is
not to predict what will happen in any specific year. Rather, the purpose of
the model is to serve as a tool for the evaluation and comparison of
alternative fishery management policies. The model provides a common methodology and
computational basis for regulation testing, and it is the comparisons between
different sets of regulations under a common set of computational assumptions
that provides fishery managers with insight into the strengths and weaknesses
of proposed regulation changes. In effect, the model serves as a computational
laboratory for regulation analysis. The challenge for the biologists and
mathematicians during the development of the model was to combine data,
equations and knowledge gained through experience into a single tool which could
provide useful quantitative information on the effects of regulation change.
The fishery model is concerned with two principal entities, salmon stocks
and salmon fisheries. It deals with the stocks of two salmon species, Chinook
and Coho. In addition to being a particular species further classified by
spawning ground location, a stock may also be thought of as a group of fish
having a common tag or mark. The term salmon fishery can refer to a collection
of individual fisheries or to a specific fishery. Each fishery in the model is
identified by the location of the fishing activity, the type of fishing gear
used, the target species of salmon, and the participants (users) in the
fishery. This is necessary since different participants can be subject to
different sets of regulations.
The fishery model contains two principal phases—calibration and
regulation analysis. The calibration phase is the major portion of the model and is
a complex process. After a calibration has been performed, however, the
resulting parameters are used to analyze alternative management policies by
multiple executions of the analysis phase. Recalibration Is necessary only
when new stock, fishery, or catch data become available.

92 FREDERICK C. JOHNSON
4. BASIC CATCH EQUATIONS
The fishery model is based on a set of equations relating to fishing and
natural mortality due to Beverton and Holt [7]. Let N(t) be the population of
a stock at time t. Then N(t) is assumed to satisfy the differential equation
N(t) - -ZN(t)
Z « M + F
where M is the natural mortality rate and F is the fishing mortality rate. Let
t be an initialization time and let N - N(t ). Then we have
-Z(t " t )
N(t) = NQe (1)
F
and C(t) - - (N - N(t)) (2)
Z °
where C(t) is the total catch in the time period (t - t ). When more than one
fishery is active, we have
F ■ I fj
j
and
FJ
Cj(t) - -- (No - N(t))
where Fj is the fishing rate in fishery j and Cj(t) is the corresponding
catch.
If we adopt a uniform time interval At ■ 1 and let N^ ■ N(t + iAt),
equations (1) and (2) can be written in the sequential form
N1+1 =■ Nte (3)
Ct - — (% - N1+1) (4)
Zi
where the mortality rates M^ and F^ are allowed to vary with i, and C^ is the
catch in the time period (t + iAt).

PRACTICAL ASPECTS OF FISHERY MANAGEMENT MODELING 93
When the time period At is small (At is one month in the model), equations
(3) and (4) can be linearized to give
Ni+1 - si Ni " Fi Ni (5)
Ct - F± Nt (6)
where S^ ■ 1 - M^ and is the survival rate. Equations (5) and (6) are the
basic catch equations of the model.
Once N , {F^} and {S^} are known, equations (5) and (6) can be used to
compute the catch of the stock in all fisheries. While {S^} can be obtained
from external studies, neither N nor {F^} are known and must be derived from a
calibration process. This holds true even for a hatchery stock for which a
relatively accurate estimate of the number of fish released is available. The
reason is that juvenile salmon enter the ocean at a very small size and
essentially 'disappear' for a year until they have attained sufficient growth to be
caught on standard fishing gear. No good estimates for natural mortality exist
for this period, but since the fish are small and more subject to predation, it
is much higher than for adult salmon.
Returning to the calibration process, let {C^} be an historical catch
record for the stock. We can include in this data set all escapement data
since they are readily available. In fact, without loss of generality,
escapement can be treated as just another fishery. Since all salmon die after
spawning, the solution to the calibration problem is to rewrite (5) and (6) in a
backwards time fashion,
Nt - (N1+1 + "C1)/S1 (7)
Fi " ^i/Ni (8>
and to initialize these equations with Nl " 0, where TL is the time period
after the oldest adult has spawned and died. Thus, the catch record and
estimates for the survival rates are adequate to derive both catch rates and the
initial stock population.

94
FREDERICK C. JOHNSON
5. KNOWNS AND UNKNOWNS
While equations (5)-(8) provide the overall framework for the model, many
important details remain to be added. First, however, we need to have a
clearer understanding of which information is known and what must be computed.
There are three categories of input data for the calibration phase of the
model:
Fishery specifications
Stock specifications
Historical catch data
The fishery specifications describe the physical and economic characteristics
of each salmon fishery, and the stock specifications describe the biological
characteristics of each salmon stock. The catch data describe the effects of
the stock-fishery interactions.
The following data are used to specify each salmon fishery:
1. area,
2. target species,
3. participant type,
4. gear type,
5. season specifications,
6. induced mortality data,
7. catch values.
The area specifies the location of the fishery, and the target species
specifies the species of a salmon caught by the fishery. The season
specifications consist of an opening date and a closing date for the fishing season
and the minimum legal length of the catch during the season. A fishery may
have more than one season during the year.
Induced mortality refers to mortality caused by the act of fishing but
which does not generate any catch value. Three types of induced mortality are
considered: hooking, drop-out and cross-species. Hooking mortality arises in

PRACTICAL ASPECTS OF FISHERY MANAGEMENT MODELING 95
a fishery which has a size-limit since the process of releasing sublegal fish
kills a significant fraction. Drop-out mortality applies to net fisheries and
occurs when legal sized fish are killed but are not landed because they drop
out of the net. Cross-species mortality arises in a mixed species situation in
which the season for one species is closed. Fishing for the open season
species causes mortality in the closed season species since these fish must be
released after landing. Induced mortality data are given as a fraction of the
catch, i.e., a 10% rate means that for each 10 legal fish caught one fish Is
killed due to induced mortality.
The catch values for commercial fisheries are ex-vessel prices in dollars
per pound (round weight). Since the dollar per pound price can change based on
the total weight of the fish, the price is specified by weight range. In other
words, a fish weighing less than 8 lbs. may be worth $0.75 per pound, while a
fish larger than 8 lbs. may be worth $1.00 per pound. A similar scheme is used
for sport fishery values, but in this case, the value is specified as dollars
per fish. Since some difference of opinion exists concerning sport fishery
values, this technique can be used to represent any particular sport valuation
philosophy.
The
1.
2.
3.
4.
5.
6.
7.
8.
stock specification data consist of:
species,
natural mortality rates,
growth data,
length frequency data,
migration paths,
substock ratios,
escapement goals,
terminal fisheries.
An underlying assumption of the fishery model is that changes in fishery
regulations will not have a significant impact on the fundamental growth rate

96 FREDERICK C. JOHNSON
of the salmon stocks. Although changes in size limit will naturally affect the
average length and weight of the catch, this is due to a change in the fraction
of the population that is legal-sized and not due to growth mechanism changes.
Therefore, the fishery model does not use a growth equation, but instead has as
input fixed monthly lengths which define the growth curve, and the data are
given for each age and maturity combination. It is assumed that the lengths of
a population are normally distributed, and the mean of the distribution is
given by the monthly length data. The standard deviation is specified by the
length frequency data which give the minimum and maximum mean lengths for the
stock and the corresponding standard deviations. Linear interpolation is used
to obtain intermediate standard deviation values. The length/weight
relationship of a stock is obtained by performing a least-squares analysis on catch
data to compute the parameters of the standard length-weight equation having
the form W - aL^, where W is weight and L is length.
While the stock Is the principal salmon entity, a stock must be further
categorized into substocks in order to account for major variations in the
migration patterns of a stock. All substocks have the same growth and natural
mortality characteristics, but differ by migration pattern. The migration data
specify the fishing areas over which a substock is distributed, and the
migration patterns are distinguished by age and maturity. The substock ratios
specify the fraction of the total stock population (at initial recruitment) for
each substock. It should be noted that these ratios will change throughout the
life of a stock due to the effects of exposure to different fisheries.
The escapement goal for a stock represents the total number of fish
required for spawning purposes. Surplus escapement or under-escapement is
automatically allocated to or taken from the specified terminal fisheries. Thus,
the model treats terminal fisheries as control fisheries which are adjusted to

PRACTICAL ASPECTS OF FISHERY MANAGEMENT MODELING
97
meet escapement goals, and this agrees with management practice. This aspect
of terminal net fishery management is a key factor in achieving a steady-state
analysis. Unless a stock is so depressed or non-terminal area fishing is so
intense that escapement goals are impossible to meet, terminal fishery
management insures a constant spawning population. This insures that the initial
population size N does not vary from year to year, a fundamental requirement
for a steady-state analysis.
The vast majority of the input data to the model consist of catch data.
The catch data consist of a number of fish caught, the average length of the
fish, and the average weight of the fish. These data are categorized by stock,
fishery, age, sex and maturity of the fish, and the time period of the catch.
These data are based on current stock size projections as applied to composites
of marked fish experiment groups and form the basis for calibrating parameters
in the model. Several hundred catch data items may be required for each
stock.
Results from recoveries of marked fish experimental groups reflect the
interactions of the stocks and fisheries under the specified fishing
regulations. Typically, the experimental data are averaged over several years to
smooth both survival and effort variations. The calibration phase thus
utilizes current stock size projections, composites of marked fish experimental
results, and the corresponding fishing regulations to calibrate the model
parameters.
It may be noted that escapement data have not appeared explicitly in the
above description. As stated earlier, escapement is treated as simply another
fishery, and the mortality for this fishery actually represents the number of
fish reaching the spawning grounds. Also, there is no explicit specification
of maturation rates for a stock. Since the catch data are distinguished by sex
and maturity, the maturation rates are derived from these data. Maturation re-

98
FREDERICK C. JOHNSON
fers to the process of attaining sexual maturity. During each year of life a
fraction of the population (different for male and female) attains sexual
maturity and begins the homeward migration. The remaining immature fish continue
the general outbound migration. Growth, natural mortality and migration
patterns are all a function of maturity.
None of the model biological processes of growth, natural mortality,
migration of a stock are affected by fishing on the stock or on other stocks, and
this is consistent with current knowledge and experience. Consequently, each
stock is a completely independent biological unit. The one mechanism which
prohibits complete independence for modeling purposes, however, is induced
mortality. This is because induced mortality is known only by species and not by
stock. This is most easily seen in the cross-species situation, but it applies
to hooking and drop-out mortality as well. Consequently, the calibration
process must allocate the induced mortality to individual stocks, and this is the
most complex task in the model.
6. THE CALIBRATION PHASE
The principal output of the calibration phase of the model consists of
initial population sizes, maturation rates and fishing rates for each stock.
The calibration process consist of four steps and utilizes a combination of
backward and forward computation.
The first step of the process is a back calculation of the catch data to
give a first estimate of the initial population for each stock. At the same
time initial estimates of the maturation rates are obtained by calculating (by
sex and age) the ratio of mature fish of the stock to the total population of
the stock. The maturation calculation is based on the January populations for
each age of the fish.
Since the catch data are distinguished by stock and not substock, the next
step is to allocate catch to substocks in order to account for differences in
migration patterns. This is a forward calculation, and the catch is allocated

PRACTICAL ASPECTS OF FISHERY MANAGEMENT MODELING 99
to each substock based upon the proportional representation of each substock in
each catch area.
After the first two steps have been completed for all stocks, the
allocation of induced mortality to substocks is the next step. Induced mortality is
allocated to all appropriate substock-age-maturity-sex combinations, and this
allocation is also based on the proportional representation of the substocks in
each catch area. As induced mortality is allocated to a substock it too is
back calculated, and final estimates of initial population and maturation rates
are obtained for each substock.
The final step of the calibration process is the computation of fishing
mortality rates for each substock. These rates are obtained by sweeping
forward in time and computing the ratio of legal catch to legal sized population
and the ratio of induced mortality to either sublegal or legal population
depending upon the type of induced mortality. All of the calibration data are
then saved for use in the regulation analysis phase.
7. THE REGULATION ANALYSIS PHASE
The catch data used in the calibration process reflect a particular set of
fishery regulations in force when the data were obtained. Thus, an analysis of
a regulation change first requires a calculation of the effects of the change
on the calibrated fishing mortality rates. Generally, a regulatory change will
increase or decrease a fishing rate as, for example, when a fishing season is
changed. In addition, there may be regulations which affect the amount of
fishing effort, or fishing effort patterns may change as a consequence of new
regulations. Such effort changes will also affect the calibrated fishing
mortality rates. The conversion of effort changes to rate changes is external to
the model and is performed by experienced fishery managers. Changes in minimum
length regulations, however, do not affect effort directly but alter the split
between the legal and sublegal portions of the stock populations. After the
regulation changes have been converted to rate changes, the simulation phase of

100 FREDERICK C. JOHNSON
the model performs a complete simulation of the stock/fishery system and
calculates the catch, catch value, and escapement obtained based on the new rates.
Input data for the regulation analysis phase of the model consist of new
fishing regulations which are to be evaluated. Only changed regulations need be
entered. In addition to regulation changes, fishery values and the number of
recruits may also be changed.
Three types of regulatory changes can be analyzed: season changes, effort
changes and minimum length changes, and these completely cover the available
flexibility of salmon fishery management. An effort change is not necessarily
accomplished by regulation, but nay be an associated change in fishing patterns
generated in response to regulatory changes. The model contains no mechanism
which attempts to predict changes in fishing patterns and levels of effort.
Instead, this is left to the judgment of the fishery manager during data
preparation.
The effects of new fishing regulations are obtained by first computing
scale factors for the calibrated fishing mortality rates and then performing a
complete life cycle simulation. For each substock population the simulation
begins with the recruitment of young fish to the fisheries. The age of
recruitment can vary, but recruitment generally occurs in the second year of
life. Based on the calibrated maturation rates for male and female fish, the
substock population is decomposed into two groups: a group of fish that will
mature during the current year of life, and a group of fish which will not
mature during the current year. The mature fish are then subjected to the
monthly processes of growth and natural mortality. At the same time, the fish
are subjected to fishing mortality. The fishing mortalities that affect the
population depend upon the migration path of the population (this determines

PRACTICAL ASPECTS OF FISHERY MANAGEMENT MODELING 101
which fisheries are actively exploiting the population), the length distribu-
bution of the population, the types of fishing gear, the regulations
controlling the fisheries, and the calibrated fishing rates.
Fishing mortality is either induced mortality and/or catch mortality. The
value of the catch is determined by the number of legal fish caught, the
average weight per fish, and the price structure of the fishery which caught
the fish. Since escapement is treated as a type of fishery, escapement
mortality actually determines the number of fish which arrive on the spawning
grounds.
The same procedure is applied to the immature fish. These fish are subject
to (generally) different growth, natural mortality and migration and,
consequently, the effects of the fisheries may be substantially different than was
the case with the mature fish. At the end of the calendar year, the remaining
immature fish undergo another maturation process, and the entire cycle Is
repeated for the fish which are now 1 year older.
The annual cycle of maturation, migration, growth, and mortality is
continued until all fish have matured and final escapement has occurred. This
completes the processing of a substock. After all of the substocks of a stock have
been processed, an escapement analysis is performed. Any surplus escapement is
allocated to the terminal fisheries or, if there is under-escapement, the
terminal fishery catch is reduced and fish are allocated to escapement. This
allocation is done on a proportional bases to maintain the correct age ratios.
After all stocks have been processed, the total catch, weight, value, and
induced mortality are known for each stock/fishery combination, and the total
escapement for each stock has been obtained. These data are summarized and
reported by the model, and this completes the regulation analysis phase.
Following a single brood year throughout its life cycle in the fisheries is, under

102 FREDERICK C. JOHNSON
steady-state conditions, equivalent to harvesting multiple age classes in a
single year. Thus, the output of the model gives a complete picture of a
single year of fishing.
8. FINAL REMARKS
Questions concerning validation and accuracy have no simple answers for
the fishery model. Appropriate historical data are not available to use as a
basis of comparison with current data for classical validation purposes.
Consequently, test and validation of the model utilized a series of simplified test
cases and parametric studies. The behavior of the model using full-scale data
has been carefully examined by management biologists and judged to be realistic
and adequate for their needs. It is simply not possible to go beyond this level
of validation at the present time. A long term goal for validation is to
evaluate the model's behavior using two different sets of steady-state catch data.
The current historic record provides one set, but fishery regulations must
stablize before a second set of data can be obtained. It is highly likely,
however, that this will not occur for several years.
The fishery model has been used by the Washington State Department of
Fisheries to analyze proposed salmon fishery regulations since 1976. Because of
the very large impact of the court mandated catch allocation and also because of
important stock conservation issues, fishing regulations have been changing very
rapidly and have received close scrutiny in both public hearings and court
proceedings. As a part of this process, the model has been used to analyze as many
as 50 different sets of new regulations and variants each year. The usefulness
of the model results from incorporating a level of detail that fully utilizes
all available data for the examination of complex and conflicting management
issues. This was possible because of the excellent understanding of available
data, biological mechanisms and management needs that was provided by the
management biologists of the Department of Fisheries.

PRACTICAL ASPECTS OF FISHERY MANAGEMENT MODELING 103
The current version of the model has the capacity to simultaneously
analyze 60 stocks, 145 substocks, 190 fisheries in 50 catch areas and 5
different age classes of catch. This represents approximately a doubling of
the capacity of the original model. The model undergoes a more or less
continuous evolution as additional features, particularly output summaries and
graphics are added. There has been no change, however, in the overall design
or philosophy of the model, and all versions have been upwards compatible. The
current version is expected to be adequate for management needs over the next
several years. A more complete discussion of the mathematics of the model may
be found in [8]. An example of the complexity of salmon management issues and
the role of the model is given in the environmental Impact statement and
fishery management plan for the 1978 fishing season [9].

104
FREDERICK C. JOHNSON
BIBLIOGRAPHY
1. P. A. Larkin, "Maybe you can't get there from here: A foreshortened
history of research in relation to management of Pacific salmon,"
J. Fisheries Res. Board of Canada, 36 (1979), 98-105.
2. C. Clark, Mathematical Bioeconoraics: The Optimal Management of
Renewable Resources, Wiley, New York, 1976.
3. , "Mathematical Models in the Economics of Renewable
Resources," SIAM Rev. 21 (1979), 81-99.
4. J. Gulland, The Management of Marine Fisheries, Univ. Washington
Press, Seattle, Washington, 1974.
5. G. Hardin, "The tragedy of the commons," Science, 162 (1968),
1243-1248.
6. R. Barsh, The Washington Fishing Rights Controversy: An Economic
Critique, Monograph Series, Univ. Washington Graduate School of
Business Administration, Seattle, Washington, 1977.
7. R. Beverton and S. Holt, "On the dynamic of exploited fish
populations," Her Majesty's Stationery Office, London, 1957.
8. F. C. Johnson, "A model for salmon fishery regulatory analysis,"
NBSIR 75-745, National Bureau of Standards, Washington, July 1975.
9. Anon. , "Final environmental impact statement and fishery management
plan for commercial and recreational salmon fisheries off the coasts
of Washington, Oregon and California commencing in 1978," U. S.
Department of Commerce, National Marine Fisheries Service, March 1978.
CENTER FOR APPLIED MATHEMATICS
NATIONAL BUREAU OF STANDARDS
WASHINGTON, D. C. 20234

Proceedings of Symposia in Applied Mathematics
Volume 25, 1981
SOME MATHEMATICAL MODELS IN HEALTH PLANNING
William P. Pierskalla
University of Pennsylvania
ABSTRACT. There are many areas in which mathematical models are
useful in health care delivery. Two of these will be discussed
here: 1) Diagnostic screening for early detection of disease,
and 2) Planning regional blood banking systems.
Non contagious diseases arise in a population in a seemingly
stochastic manner. If testing procedures exist which are capable
of detecting the disease before it would otherwise become known,
and if such early detection provides benefit, the periodic
administration of such a test procedure to the members of the
population, that is, a mass screening program, may be advisable.
Moreover, if the population is composed of sub-populations which
exhibit different disease incidence rates and different unit costs
of test applications, and if different tests which have different
reliabilities for detecting the disease are available, then the
question of allocating limited screening resources among the sub-
populations arises. The optimal allocation depends upon the form
of the disutility functions of the sub-populations. Comprehensive
analytic models are needed to perform this allocation.
Health planning can be viewed from many perspectives. Perhaps
the most critical one facing the United States today is to contain
the costs of health care and yet deliver quality care to the entire
population of the U.S. Certain aspects of planning to achieve
these objectives must be undertaken on a regional level, others
at a sub-regional level, and still others at the institutional
level. An integrated hierarchy of analytical models is needed to
link the decisions at each of these levels. Decisions at the
macro level involve the appropriate numbers of people by skills,
numbers of facilities, and technological sophistication for a
region. At the middle level, the decisions involve facility
locations, their levels of technology and services and personnel needs
to achieve minimum cost yet provide accessibility and quality of
care in the sub-regions. At the institutional or micro level,
analytic models are used to determine admissions and appointments,
inventory levels and capital equipment, daily and weekly staffing,
and facility scheduling. These different levels of modeling are
illustrated in the context of planning for regional blood-bank
systems.
1980 Mathematics Subject Classification 90-02, 90B05, 90B25.
This work was partially supported by the National Science Foundation
under grant ENG77-07463 and by Office of Naval Research Contract
N00014-75-C-0797 (Task NR042-322).
Copyright © 1981 American Mathematical Society
105

I. DIAGNOSTIC SCREENING FOR EARLY DETECTION OF DISEASE
1. INTRODUCTION
There are many situations where a defect can occur randomly among the
members of a population -- either a population of human beings, inanimate
objects, or perhaps livestock -- and once present, exist and develop, at
least for a time, without any manifest symptoms. If the early detection of
such a defect provides benefit, it may be worthwhile to employ a test
capable of revealing the defect's existence in its earlier stages. (A defect,
disorder, or disease will generally be referred to simply as a defect; and
the word unit or individual will refer to a member of the population.)
Of course, continuous monitoring would provide the most immediate such
revelation. But considerations of expense and practicality will frequently
rule out continuous monitoring so that a schedule of periodic testing -- a
screening program -- may be the most practical means of achieving early
detection of the defect. In general terms, the question then becomes one
of how best to trade off the expense of testing against the benefits to be
achieved from detecting the defect in an earlier stage of development. The
expense of testing increases both with the frequency of test applications
and with the cost of the type of test used. The benefits increase with the
frequency and the quality of test used and the quality of a test is often
directly related to its cost.
The benefits of early detection also often depend upon the application
considered. For example, in a human population being screened for some
chronic disease (cancer, glaucoma, heart disease, etc.) the benefits of
early detection might include an improved probability of ultimate cure,
diminished time period of disability, discomfort, and loss of earnings, and
reduced treatment cost. If the population being screened consists of
machines engaged in some kind of production, the benefits of early detection
might include a less costly ultimate repair and a reduction in the time
period during which a faulty product is being unknowingly produced. If the
population being screened consists of machines held in readiness to meet
some emergency situation, an early detection of a defect would reduce the
the time the machine was not serving its protective function. This process
106

SOME MATHEMATICAL MODELS IN HEALTH PLANNING
107
of inspecting a sizable population for defects is called mass screening.
The expense of testing includes easily quantifiable economic costs such
as those of the labor and materials needed to administer the testing.
However, there can also be other important cost components which are more
difficult to quantify. For example, in the case of a human population subject to
medical screening, the cost of testing includes the inconvenience and possible
discomfort necessitated by the test; the cost of false positives which
entails both emotional distress and the need to do unnecessary follow-up
testing; and even the risk of physical harm to the testee, e.g., from the
cumulative effect of X-ray exposure.
The rationale for constructing a mathematical model of mass screening
is to provide a conceptual framework within which a mass screening program
might best be designed and its worth evaluated. Such a design must Jeter-
mine which kind of testing technology will be used. Several candidate
technologies may be available, each with different reliability
characteristics and costs. In addition, the frequency of testing must be decided.
If the target population can be partitioned into sub-populations according
to susceptibility to the defect (e.g., by age, family background, time since
last overhaul (for a machine), etc.), then the best allocation of the
testing budget among the sub-populations must be determined. Lastly, a decision
must be made, for the population and type of defect under consideration,
whether a mass screening program is justified at all. It is felt that the
above determinations are best carried out within the conceptual framework
of a cogent model of mass screening.
In the following sections a reasonably comprehensive model for
decision making with regard to mass screening is presented. The model utilizes
a time-based approach. But rather than seeking to analyze or minimize
detection delay, as in some of the literature on this topic, the objective
function is an arbitrary increasing function of detection delay.
(Detection delay is the time between the incidence of the defect and its detection,
regardless of whether that detection is the result of a screening test or
of the defect becoming self-evident.) The reason for choosing a general
function is that the disutility experienced upon the delayed detection of
a defect may well vary in a highly nonlinear way with the length of the
delay.
Another objective sometimes used for inspection models is to maximize
the lead time where the lead time is defined as the difference between the
time of detection via a screening test and the time detection would
otherwise have occurred had not a screening program been in existence. However,
in the case in which the test being used is perfectly reliable then it be-

108
William P. Pierskalla
comes possible to consider the lead-time criterion in terms of the
following simple function of detection delay. Suppose that at the (possibly
random) age T (measured from the time of incidence of the defect) the
defect, in the normal course of its development, would become manifest even
without a screening test. Then, if the defect is detected at age t, the
lead-time gained is (T - t) . Therefore, if sup T is finite, the function
D(t) = sup T - E(T-t) may be used as a disutility function which, when
minimized, will maximize the expected lead-time.
Consequently, the results discussed here based on an arbitrary
disutility function, generalize as well as extend results in earlier work. In
addition, new results are presented.
In the next section a brief review of the literature is given. Since
the work on inspection models is very large, only the most relevant papers
are discussed. The interested reader may refer to the surveys by McCall
[1965] and Pierskalla and Voelker [1976] for a more comprehensive review.
In the third section the model is formally stated and an expression
derived for the expected disutility per unit time incurred under a regime of
uniformly spaced test applications.
This model can be specialized to the case of a perfect test; i.e.,
when the test is administered to an individual with the defect, the defect
will be detected with certainty. Under this assumption, a regime of
uniform test intervals is optimal within a wider class of "cyclic" testing
schedules for a single sub-population. The problem of allocating a
screening budget among various sub-populations can then be analyzed. As part of
this analysis, the explicit screening schedule (r , r , ..., r0), where r.
designates the optimal testing frequency for members of the jth
sub-population, can be obtained in terms of the sub-population-specific incidence
rates N.X. and the budget constraint for the disutility function D(t) = at .
For D(-) convex, the above solution (when m = 1) provides a bound on the
ratios r./r..
Other uses of the model involve the analysis of the case when the
probability of detection is a constant over all values of elapsed time since
incidence. The long-run expected disutility per unit time can be derived;
its differential qualities (with respect to variations in the test
reliability parameter and testing frequency) exhibited; and explicit solutions
provided for various special forms of D(-)- Also for special forms of
D(-)> decision rules can be presented to select between two different kinds
The function u = max(u, o).

SOME MATHEMATICAL MODELS IN HEALTH PLANNING
109
of tests which differ with respect to their reliabilities and cost per
application.
If it is assumed that the test will detect the defect if and only
if the elapsed time since incidence exceeds (or equals) a critical threshold T
which characterizes the test, an expression for the long-run expected
disutility per unit time can be derived. Then decision rules can be developed to
select between two alternative test types which differ with respect to their
critical threshold T and their cost per application. Some interesting examples
for linear and quadratic disutility are given in the fourth section.
The last section in Part I provides a technique to estimate the shape of
the disutility function from empirical data in the case of a perfect test.
Finally, it should be mentioned that the population (or each
sub-population, in the instances where a heterogeneous population is considered) is
assumed to be of fixed size, N, and the defect to arise according to a
stationary Poisson process with rate NX (N could be a very large but finite
number). It may be somewhat more realistic to set the defect occurrence rate
(sometimes called the defect arrival rate) proportional to the number of
defect-free units, rather than to the total number of units in the population.
However, it is also assumed that no defect can remain undetected longer than
T , even without any screening tests being given. Hence, T NX is an upper
bound on the expected number of undetected defects in the population. If it is
the case that once a defect is detected, the afflicted unit is replaced in the
*
population with a healthy unit, then T NX represents a bound on the expected
difference between the number of healthy units and the total number of units in
the population. For X small, as would be the case for a relatively
infrequently occurring disorder, this should represent no difficulty. Consequently, it
is assumed that X is small relative to N, which is consistent with the examples
of potential applications which have been or will be mentioned.
2. LITERATURE REVIEW
Some of the early papers which have a bearing on time dependent models of
mass screening are: Derman [1961], Roeloffs [1963, 1967], Barlow, Hunter, and
Proschan [1963] and Keller [1974]. Kirch and Klein [1974], whose paper is
addressed explicitly to a mass screening application, seek an inspection
schedule which will minimize expected direction delay subject
to a constraint on the expected number of examinations an individual would
incur over a lifetime; the test is assumed perfectly reliable. The point of
view adopted here in Part I is similar to that of Kirch and Klein. However,
one respect in which the approach here differs from that of Kirch and Klein is
that several sub-populations are considered each with its own characteristic

110
William P. Pierskalla
incidence rate for the disorder. The idea is then to allocate optimally a
fixed screening budget among the sub-populations. Kirch and Klein instead
take a longitudinal view. An individual through his lifetime is subject to
different probabilities of incurring the defect and a screening schedule is
optimized subject to a constraint on the expected number of examinations over
a lifetime.
McCall [1969] considered the problem of scheduling dental examinations
under the assumption that the time between the incidence of a cavity and the
scheduled dental examination controls whether the cavity results in a filling
or an extraction. Cavities are assumed to occur according to a Poisson
process. As a generalization, he permits the time required for a cavity to
become beyond repair (by a filling) to be a random variable.
Lincoln and Weiss [1964] studied the statistical characteristics of
detection delay under the assumption that the times of examinations form a renewal
process and that the probability of detecting the defect, p(t), is a function
of the defect's age, t. They derive equations, similar to renewal type
equations, which relate the density functions for the following entities: the
probability of detection at a test application (p(t)), the time until the
defect becomes potentially detectable and from this time the forward recurrence
time to the first test, the probability of the event that at a particular time
a test occurs and all prior tests had failed to detect the defect, and the
detection delay. For the two special cases where p(t) is a constant and where
p(t) is exponential, the moments for the detection delay are derived in closed
form. For uniform testing intervals (and general p(-))> the distribution and
moments of the detection delay are computed. For p(t) = 1, they solve for that
testing schedule which maximizes the time between tests subject to a constraint
on the performance of the screening program relative to detection delay. Two
such constraints are considered. The first bounds the probability of
detection delay exceeding some threshold T. The second bounds the mean detection
delay.
theorem 1.3.1 in the next section, which gives the expected disutility
in terms of the test interval and test type, is similar to the objective
function given by Lincoln and Weiss although their approach, as outlined above,
is different.
Several recent papers have studied the statistical characteristics of the
lead-time provided by a screening program -- either a one-shot screen of the
population or a periodic screening program. Some of these papers are:
Hutchison and Shapiro [1968], Zelen [1971], and Prorok [1973].
In some recent work, Eddy [1978a, 1978b] describes a semi-Markov Chain
analysis for the screening and detection of cancer. This modeling effort was

SOME MATHEMATICAL MODELS IN HEALTH PLANNING
111
undertaken to devise a practical planning tool based in the reality of a
breast cancer screening program.
Finally a few authors, Schwartz and Galliher [1975], Thompson and Disney
[1976] and Voelker [1976] let both the reliability of the test and the
disutility (or utility) of detection be a function of the defectfs state rather
than of time since the defect's incidence. Although such models are more
general and do utilize a general concept of disutility, they have not been
amenable to closed form evaluation of expected disutility.
To incorporate random defect occurrences into their models, previous
researchers focus upon an individual who will incur the defect. They use the
density function for the age when that individual incurs the defect as a
fundamental element of their model. Since the density function reflects age-
specific incidence rates, a "life time" testing schedule can, thereby, be
developed to tailor testing frequency at each age to the probability that the
defect will occur at the age.
Our way of modeling the randomness of these occurrences reflects a
somewhat different perspective on the mass screening problem. We look through the
eyes of a decision-maker charged with intelligently allocating a fixed budget.
The time frame over which the allocation must be made is often short compared
to a typical life time of a member of the client population. Therefore, the
decision-maker does not plan lifetime screening schedules for particular
individuals. Instead, he tries to maximize the benefit that can be derived
from his available budget over a much shorter planning horizon.
With the problem viewed in this perspective, the random nature of defect
occurrences is most naturally modeled as a Poisson process with its parameter
determined by the incidence rate of the defect and the size of the population.
This approach has proved particularly useful in the following context: if
different segments of the client population exhibit different incidence rates,
sub-populations can be defined with defect incidence within each of them
modeled as a Poisson process with its respective parameter. Then the budget
can be so allocated among the sub-populations as to permit appropriate
relative testing frequencies (cf. Voelker and Pierskalla [1978]). In this way,
age-specific incidence rates can be incorporated into the notion of Poisson
defect occurrences. Moreover, factors other than age which affect defect
incidence rates (family history, smoking habits, work environment, etc.) can
also be incorporated into the model.
3. A MODEL FOR MASS SCREENING
Although most of the definitions are stated prior to their use in each
section, listed below are some notation and conventions used as various times

112
William P. Pierskalla
throughout the paper.
1 (t) = 1 (t) = <
UJ r Lr> r J
1 if » < t <Hil
r — — r
0 otherwise
(2) II x. = 1 for n < 1
n
n
i=l
n
i=l
X.
i
X.
1
=
1
0
(3) I x. = 0 for n < 1
(4) [x] is the largest integer not exceeding x.
(5) The phrase "increasing function" shall mean a strictly increasing
function.
As mentioned previously, the objective function will not be to maximize
expected utility, but rather to minimize expected disutility. The disutility
associated with a particular detection will depend only on the elapsed time
since the defect's incidence. The notation D(t) will express the disutility
incurred if detection occurs t units of time after incidence. D(-) is assumed
throughout this paper to be a nonncgative increasing function.
In those cases where sup D(s) < °°, there is a natural relation between
s _> 0
the notion of a utility function and D(-); namely,
U(t) = sup D(s) - D(t)
s > 0
U(t) is a decreasing function expressing the disutility avoided by detecting a
defect t units of time after its incidence.
Initially, only a single susceptibility class of size N will be
considered. The results will then be generalized to an arbitrary number of
subclasses. The times of incidence for the defect in the population are assumed
to form a Poisson process with parameter NX and are designated by the sequence
k
{S }, k = 1, 2,... This assumption is motivated by the fact that any
occurrence or arrival process with the following characteristics is a Poisson
process: at the time of an arrival there is almost surely (i.e., with probability
one) only one arrival, that the number of arrivals in a time interval does not
depend on past arrivals, and that the number of arrivals in intervals of equal
length are identically distributed (Cinlar [1975]). For many applications
such as the occurrence of diseases like cancer, heart trouble, etc., this

SOME MATHEMATICAL MODELS IN HEALTH PLANNING
113
assumption is quite reasonable.
Were a screening test of type I to be administered to the individual with
kth defect (i.e., kth in the order of
then the test outcome random variable is:
the kth defect (i.e., kth in the order of defect incidence) at time S + t,
DEFINITION:
Y\(t) = *
if kth defect is not detected at time t
after incidence
if kth defect is detected at time t after
incidence.
(3.1)
Since a defect cannot be detected before its incidence, Y0(t) = 0 for t < 0.
k
Notice that the argument of Yc(-) refers to time relative to the incidence of
the defect. If the population is screened at time s, where s is absolute
time, then the kth defect will be detected if and only if S £ s and
k k
Y*(s - SK) = 1.
k i
It is assumed that Y.(t), Y^(s) are independent except when k = j and
t = s, and that P(Y?(t) = 1) depends only on I and t. This latter assumption
makes possible the
DEFINITIONi
Pfc(t) = P(Yj(t) = 1). (3.2)
The function p-(0 describes the reliability characteristics of the type
I screening test, i.e., how likely such a test is to detect a defect as a
function of the defect's age. Note that P«(t) = 0 for t < 0.
In this section the screening test is assumed to be administered to the
entire population at the titoes 1/r, 2/r, 3/r,... The testing frequency r is
a control variable. In the next section, upon assuming perfect test
reliability, it is shown that the above schedule of uniform testing intervals is
optimal within a wider class of "cyclic" schedules.
The random variable S „ denotes the time at which the kth defect is
-k r,£ k
detected. S . depends on the arrival time of the defect (S ), the type of
test used (&), and the testing frequency (r).
DEFINITION:
S* = min tn/r|Yj(2. - Sk) = 1} (3.3)
n=l,2,...

114
William P. Pierskalla
Given the application of test type I at the times {l/r, 2/r,...}, the
-k k
disutility incurred by the kth defect is D(S 0 - 5 ). The total disutility
incurred due to those defects which occurred in the interval [j/r, (j+l)/r)],
when test I is used is:
DEFINITION:
oo
B 0 . = I D(Sk 0 - Sk)l-/ (S ).
r* ,J k=l r> j/
The following proposition provides an expression for E[B . .] in terms
r ,36, j
only of the disutility due to detection delay (as expressed by D(-)) and of
the reliability of test type I (as expressed by p.(•))• In addition, the
theorem shows that E[B 0 .] = E[B 0 n] for j = 0, 1, 2,...
r,X/,j r,X/,u
THEOREM 1.3.1:
n
~ r n-1
E[B ] = NX I / D(u)p£(u) n [l-pA(u-^)]
' ,J n=l , m=l
?)]du
n-1
r
for j = 0, 1, 2,...
This proposition yields a relatively simple expression for the expected
total disutility in any interval of length l/r and using test type I with
probability of detection Pn(*)- This expectation is used in the objective
function of a mathematical program to determine the optimal testing frequency for
a mass screening program for a heterogeneous population. In order to develop
this mathematical program, the following definitions are useful.
DEFINITION:
n-1
B 0 = lim - I E[B 0 .].
T,i ^_ n . _ L r,£,iJ
B 0 is the long-run expected disutility per unit time given the testing fre-
r ,X/
quency r and the test type I. (The factor r enters the definition to convert
disutility per unit testing-interval into per unit time.) By Theorem 1.3.1,
E[Br,£,j] = E[Br,£,0] for J = °> X> 2>"- Therefore,

SOME MATHEMATICAL MODELS IN HEALTH PLANNING
115
\,SL = ^r,*^
11
00 r n-1
= rNX I / D(u)p£(u) n [1 - p£(u - ^)]du. (3.4)
n=l - m=l
n-1
Notice that if test type I provides perfect reliability, i.e., pp(t) = 1
for t > 0, then (3.4) gives
B 0 = rN X J D(u)du. (3.5)
Now consider the problem of selecting testing frequencies and test-types
for each of Q different susceptibility classes which together comprise the
whole population. These classes may differ from one another in the number of
units they contain, in their defect incidence intensity, and in the cost per
test application to an individual for a particular type of test. The sub-
populations, however, are assumed to share a common D(-) function.
Using (3.4) the expected long-run disutility per unit time for
sub-population j with frequency r(j) and test £(j) where j = 1, 2,...,Q is given by:
i
h.Tit)M» - r(j)V(j) Jj /r0) D(u)p*u)(u) 3| [1-p*o)(u - F&y)]du-
i-1
r(j)
This expression can be used to formulate a multi-sub-population screening
problem subject to a budget constraint as follows:
Minimize n
(r(l),...,r(Q), A(l),...,*(<») I *j ,r(j) ,A(j) (3*6)
such that
^Njcj,*(j)rjib (3-7)
r. > 0 j=l,...,Q (3.8)
A(j) erf j=l,...,Q (3.9)

116
William P. Pierskalla
where
c. pr.. = cost per application of a test of type £(j) to an
^' U individual of sub-population j,
N. = number of units (or individuals) in sub-population j,
b = budget per unit time.
J^ - set of all feasible tests .
In order to make this mathematical program even more comprehensive, it
is possible to add constraints on the amount of testing labor available and on
the capacity of the testing facilities in terms of the number of arrivals, the
frequency of testing and the type of tests used. In addition, the cost of
false positives can be included as a part of the test costs in inequality
(3.7).
For example, a constraint on the total labor available is:
£ NjV(j)rj±Li (3ao)
and on the total testing facilities available is:
for each type of test £(•) used over the sub-populations being tested, where
6. . . = amount of labor needed to administer test type £(j)
•* * J to an individual of population j,
f. «,.. = amount of testing facility time needed to administer
J* ^J test type £(j) to an individual of sub-population j,
L« = total amount of labor available to administer test
type &(•) per unit time,
F- = total amount of facility time available to administer
test type £(•) per unit time.
4. SOME EXAMPLES OF DISUTILITY FUNCTIONS
Suppose a production process is subject to a randomly occurring defect.
Although production appears to proceed normally after the incidence of the
defect, the product produced is, thereafter, defective to an extent which
remains constant until the production process is returned to its proper mode

SOME MATHEMATICAL MODELS IN HEALTH PLANNING
117
of operation. The only way to learn if the production process is in this
degraded state is to perform a costly test. Now, if a test detects the
existence of the degraded mode of production t units of time after its
incidence, the harm done will be proportional to the amount of defection
product (unknowingly) produced which, in turn, is proportional to t. Hence,
D(t) = at for some a > 0.
Another example where a linear D(-) function may be appropriate would be
for the periodic inspection of an inactive device (such as a missile) stored
for possible use in an emergency. If t is the time between the incidence of
the disorder and its detection, the disutility incurred is proportional to the
probability that the device would be needed in that time interval. If such
"emergencies" arise according to a Poisson process with rate y, then the
probability of an emergency in a time interval of length t is 1 - e , which,
for y small, is approximately ut. Hence, if b is the cost incurred should
there be an emergency while the device is defective, and if y is the (small)
arrival rate of emergencies, then D(t) = byt.
A quadratic disutility could arise in the following situation. Suppose
the magnitude of a randomly occurring defect increases linearly with time
since the occurrence of the defect. For example, the magnitude of the defect
might be the size of a small leak in a storage container for a fluid, and as
fluid escapes, the leak gets larger. Further, suppose that the harm done
accumulates at a rate proportional to the magnitude of the defect. Hence, the
quantity of fluid lost (at least initially) increases the longer the defect
exists, and the rate of fluid loss is proportional to the size of the leak.
Let the size of the leak (as measured by rate of fluid loss), at a time
s since the leak's incidence, be cs. Then, if the defect is detected at time
t since incidence, the disutility incurred (fluid lost) is D(t) = /n csds
2
= 1/2 ct
5. EMPIRICAL ESTIMATION OF D(-)
Although data may not be available currently to support the utilization
of particular models, that should not deter the development of such models.
It is reasonable to expect that in many application areas in the future much
additional data will become available; for example, more data will become
available concerning the stochastic pattern of a particular disease's
development. Furthermore, a good model will serve as a guide to the kinds of data
which should be gathered.
It is also reasonable to expect the future development of improved
testing technologies capable of detecting a disorder in a much earlier stage of
development than is now the case. Such innovations will make screening

118
William P. Pierskalla
programs more attractive and, consequently, make more important the analytical
tools to design such programs intelligently.
However, it is possible to provide a simple procedure to estimate
empirically the disutility function D(-) under the assumption of perfect
detection. Such a procedure is necessary, since upon the detection of a
disorder there may be no way of directly determining how long the disorder has
been present, although that length of time could not exceed x = —. It is
assumed, however, that at each detection of a defect, the degree of disutility
incurred due to that defect can be observed. For example, in the case of
medical screening, at the detection of a tumor its degree of development can
be noted even if it is not possible to determine exactly when, since the last
test, the tumor originated.
In order to allocate optimally a screening budget among differing sub-
populations (susceptibility classes), as was discussed in Section 3, it is
necessary to know the shape of the disutility function. Under the reasonable
assumption that D(-) is an increasing continuous function, the function may be
derived in the following manner. For a particular population subject to a
k
Poisson defect arrival iS } with rate NX, arbitrarily select a value for x
and set up a prototype mass screening program for the population in which
tests are made (using a perfect test) at times 0, x, 2x,... The manner of
defects detected at the time jx, which are characterized by a disutility less
than or equal to y, record and designate by Q.(y). Then,
Vy) = j0Vy](D(jx-sk)) ^jx-x. jx]<sk>' J-1.2.S.-
Q.(y) is observable for all values of disutility y > 0. Further, Q.(y) and
Q.(y) are independent and indentically distributed random variables for i^i
and y > 0. Therefore, by the law of large numbers,
1 n
lim ± Z Q.(y) = E[Q (y)] for all y > 0.
n-K» n j = i J
Hence, the function y -* E[Q (y)] may be estimated by collecting a
sufficiently large sample of the functions Q , Q?,...
Let H(y) = E[Q (y)]. Once H(-) is estimated in this manner, the
following theorem shows how the desired function D(-) may be obtained from H(-)>

SOME MATHEMATICAL MODELS IN HEALTH PLANNING
119
THEOREM 1.3.2: If D(0) = 0 and D(-) is continuous and increasing on [0, x].
then
D(t) = H_1(NXt) for 0 <_ t <_ x .
By observing the Q.(y) and taking the inverse of their expectation, an
estimate for D(t) may be obtained. For some diseases such as heart disease,
glaucoma, and some cancers, there may be enough data currently available to
begin estimating the disutility functions and to start computing optimal
testing frequencies and tests.
II. PLANNING REGIONAL BLOOD BANKING SYSTEMS
1. A HIERARCHY OF PLANNING MODELS
Although health planning involves many activities and mathematical
modeling aspects, we will only discuss hierarchies of models to link decisions at
the regional, sub-regional and institutional levels. Decisions at the macro
level involve the appropriate numbers of people by skills, numbers of
facilities, and technological sophistication for a region. At the middle level,
the decisions involve facility locations, their levels of technology and
services and personnel needs to achieve minimum cost yet provide accessibility and
quality of care in the sub-regions. At the institutional or micro level,
analytic models are used to determine admissions and appointments, inventory
levels and capital equipment, daily and weekly staffing, and facility
scheduling. At all levels it is necessary to make these decisions by trading off
competing objectives such as minimizing costs, increasing quality of care, and
increasing accessibility and availability of services.
Rather than discuss these modeling activities in an abstract form, which
could easily be done, we will focus on regionalization in blood banking.
Blood banks are an important and integral part of health service systems.
2
Their main functions are blood procurement, processing, cross-matching,
storage, distribution, recycling, pricing, quality control and outdating. The
large blood banks are often also responsible for blood research, disease and
Cross-matching is the procedure of testing the donor's blood with a sample of
blood from a potential recipient (patient) to determine whether the two types
of blood are compatible and therefore will not lead to medical complications
wnen tne patient is transfused.

120
William P. Pierskalla
reaction prevention. In recent years, there has been much discussion on the
issue of regionalization of blood banking systems, in the hope of decreasing
shortages, outdates and operating costs, without sacrificing blood quality,
research and education.
In a broad sense, regionalization is a process by which blood banks
within a given geographical area move toward the coordination of their activities.
Such coordination may range from cases in which the blood banks merge into a
large, centralized unit, to cases where the existing structure remains
unaltered and only certain functions, such as donor recruitment, processing and
distribution, are coordinated among the blood banks. In most of these cases,
questions of optimal region size, central and local bank locations, regional
boundaries, optimal distribution and communication network configurations must
be answered. Also, administrative policies, ordering and cross-matching
policies, and donor recruitment and component therapy strategies must be
analyzed and coordinated.
In an earlier paper on regionalization, (Cohen [1975]), hierarchical
structures for different types of regional blood banking systems were discussed.
The appropriate structure for any area depends on a complex interaction
between the level of activity, the economies or diseconomies of scale, the cost
effectiveness and efficiency, as well as the interactions of the interested
parties. In this section, the impact of size and structure on the different
costs of operation of blood banking activities in a region will be analyzed.
The main determinants of cost are the personnel, the space, the equipment,
the location and allocation of facilities and the transportation used to carry
out the activities of the blood center.
Since regions vary in terms of their geography, number of donors, and
number of recipients of blood services, the most effective and efficient
organizational structure will also vary from region to region. It is not our
purpose here to delve into all the many different ramifications of the
different regional structures. Rather for this analysis it is sufficient to
consider three generic structures of regional blood banking in the United
States. Most systems which do not exactly fit into one of these structures
can be closely approximated by one of them. The structures are given in
Figure 1, and represent (i) a community blood center which services the entire
needs of a particular region, (ii) a collection of (communicating) community
blood centers each under its own independent control, which services the blood
needs of a particular region, and (iii) a regional blood center which either
coordinates or controls the activities of a collection of community blood
centers which in turn fulfill the needs of the community in the region. As a
general rule, as the size of a region changes due to an increase in demand or

SOME MATHEMATICAL MODELS IN HEALTH PLANNING
121
CENTRALIZATION - -COORDINATION MODELS
TS TS TS TS
TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS TS
RBC = REGIONAL BLOOD CENTER
CBC = COMMUNITY BLOOD CENTER
TS = TRANSFUSION SERVICE
Centralization may involve: Coordination may involve:
• Power Transfer • Cooperative Donor Recruitment
• Centralized Information System • Resource Sharing
• Mergers, Relocations • Distribution and Processing
» Share Information
• Coordinate Planning
Figure 1: Regional Blood Banking

122
William P. Pierskalla
geography, the single community blood center may not adequately fulfill the
needs of the region and one of the other two structures tends to replace the
single center over time. In some cases, the new structure consists of a
regional community blood center with satellite facilities.
In the next section, we focus on the location-allocation-transportation
aspects of regionalization. Our variables will be central bank locations,
regional boundaries and blood distribution network configurations. In the
first model, all the other aspects of regionalization are summarized by
certain terms called system costs, which are primarily functions of two
factors: the number of hospitals in a region and the amount of blood used by
each hospital in the region. Both of these factors are functionally related
to the variables considered in the location-allocation-transportation model
(since they vary with the regional boundaries.) The other factors that affect
the non-transportation aspects of regionalization tend to be independent of
the variables in the model, consequently they are independent of the location-
allocation-transportation decisions.
The third section describes some middle level and institutional level
models in their abstract forms. Although these mathematical models are
directly applicable to the blood bank decision process, they do provide qualitative
insights into the considerations underlying the optimal decisions and the
general structure of such decisions.
The final section briefly summarizes the findings.
2. THE REGIONAL MODEL AND DECISIONS
The regionalization model is verbally described as follows: "Within a
given geographical area, there are N hospitals. Regionalization is to be
achieved by dividing the area into M regions and establishing a central blood
bank in each region. All blood banking activities in a region are to be
coordinated. Supply generation is to be done mainly by each central bank and
each hospital is to obtain its primary blood supply from the central bank in
its region. The blood distribution operation consists of periodic and
emergency deliveries. The hospitals in a region receive their periodic daily
requirements from their central bank. The blood deliveries are made by
vehicles which, starting from the central bank, visit one by one the hospitals
they are scheduled to supply, and return to the central bank. These vehicles
have given capacities and given limits on the number of deliveries they can
make per day. Because of the wide fluctuations in demand, a hospital may
deplete certain blood types before the next periodic delivery is due. In that
case, a delivery vehicle is dispatched immediately, from its central bank.

SOME MATHEMATICAL MODELS IN HEALTH PLANNING
123
The delivery vehicle makes an emergency blood delivery to that hospital and
returns to the blood bank. The problem is to decide how many central blood
banks to set up, where to locate them, how to allocate the hospitals to the
banks, and how to route the periodic supply operation, so that the total of
transportation costs (periodic and emergency supply costs) and the other system
costs are a minimum." This problem will be called the Blood Transportation-
Allocation Problem (BTAP).
In modeling the BTAP it is necessary to describe the costs as functions
of the decision variables. The periodic delivery costs, which are a set of
linear terms, depend on all three types of variables in this problem, that is,
routing the periodic supply operations, locating the banks, and allocating
the hospitals to the banks. The emergency costs are also a set of linear terms
and depend on only two types of variables, locating the banks and allocating
the hospitals. The system costs are nonlinear functions of the size of the
blood banks and the number of hospitals allocated to the blood banks, and
therefore, of all the variables in this problem they depend only on hospital
allocations. So, if the system costs were constant and the emergency costs
were negligible, the model would be equivalent to the General Transportation
Problem (GTP) (see Or [1976], or Magnanti et al. [1975]). If the system
costs were constant and the periodic delivery costs were negligible, the model
would reduce to a Location-Allocation problem (LAP) (see Cooper [1963, 1964],
Hurter and Wendell [1973a, 1973b], or Francis and White [1974]). So the model
is a complex combination of these two large problems. The basic strategy we
use in order to obtain a good solution depends heavily on these two sub-
problems. We solve each subproblem independently and then combine them at the
end, making tradeoffs between them and superimposing the system costs
considerations, to obtain a good solution to the model. However, it should be noted
that, unlike the GTP, solving the LAP does not produce a complete, feasible
solution to the main model. It only gives the locations and the allocations,
and in order to get the missing periodic delivery routes, one must solve a set
of vehicle dispatch problems.
Other work on regionalization or centralization of blood bank activities
does not consider the location-allocation-transportation aspects. Jennings
[1970, 1972] used a simulation model to construct part of a regional blood
banking system. He grouped a number of identical hospitals together; however,
he did not have a central blood bank. Transshipment policies and inventory
levels were studied to see their impact on shortages and outdates. Yen [1965]
also studied multiechelon inventory systems. He concentrated his efforts on
the optimal inventory levels and the optimal issuing policies. Prastacos
[1977] and Prastacos et al. [1977] were interested in the allocation of exist-

124
William P. Pierskalla
ing stocks among the hospitals. Neither Jennings, Yen nor Prastacos et al.
studied the location of central banks or the allocation of hospitals to them.
THE BLOOD TRANSPORTATION-ALLOCATION MODEL
Because the BTAP is a complex optimization problem, we will make a few
reasonable assumptions to decompose the problem into smaller subproblems.
ASSUMPTION 1: The number of banks, M, is a given number.
Even in cases in which the above assumption doesn't hold, M is almost always
restricted to a small, finite, feasible set (M is always an integer and
1 £ M <_ N). So, in those cases one could solve the problem for each feasible
value of M to get the optimal solution. In this respect, Assumption 1 is not
restrictive.
ASSUMPTION 2: The blood delivery period is daily for each hospital.
Considering that some hospitals use more than 7000 units of blood per year,
while some others use less than 10, this is an unrealistic assumption.
Unfortunately, determining the optimal multiple delivery periods as well as
the location-allocation and routings increases the complexity of the problem
considerably and makes it almost impossible to find a direct solution
procedure. A simple, multiple period problem should have three options for the
periodic deliveries (daily, biweekly, weekly). However, the problem of
choosing optimal periods for each hospital would be a very large (and time
consuming) combinatoric process. If the periods are set in advance (daily,
biweekly, weekly) these multiple periods can be easily incorporated into the
present location-allocation model merely by adjusting the costs to reflect
costs per day. The routes of the delivery vehicles however would have to be
adjusted later.
ASSUMPTION 3: The potential locations of the M banks are a finite (and
usually small) number.
The set of practical feasible locations is almost always a small, finite
set (usually one does not want to build a blood bank from scratch; instead,
they are most frequently located in the area's largest hospitals or at
existing blood centers). So, if necessary, we could solve the problem for all

SOME MATHEMATICAL MODELS IN HEALTH PLANNING
125
combinations of feasible locations, to get the optimal solution. In a design
problem, this is not an impossible enumeration, since even a region the size
of the Chicago metropolitan area (using the amount of blood transfused as a
measure of hospital size) has only six central blood banks and has only seven
hospitals with consumption rates of over 7000 units per year.
Finally, it is assumed that each vehicle makes one (non-emergency) trip
per day. If multiple trips per day are allowed in a real setting this model
would need to be modified appropriately. Most of the latter do not qualify
to be central blood banks for various reasons. So, in this respect Assumption
J. is not very restrictive.
The following notation will be used in formulating the BTAP.
i) N is the number of demand points (i.e., hospitals),
ii) M is the number of supply points (i.e., banks)
iii) n is the maximum number of supply vehicles available,
iv) ^ = {H ,...,H } is a set of N demand points.
v) A - {H ,. ..,R. } is a set of M supply points.
vi) ^ = tf\Js is the set of all points involved in the problem.
vii) d.. is the "distance" from H. to H.. It should be noted that
although Euclidean distances among locations of hospitals and
central banks are used in the solution procedure, one could
obtain a matrix of accurate travel times between all pairs
of hospitals and banks, and one could use this matrix or any
other "distance measure" instead of the Euclidean distance
matrix.
viii) C, , k = l,...,n is the capacity of supply vehicle k.
ix) Q., i = 1,...,N is the requirement of demand point i.
x) D, , k = l,...,n is the maximum distance supply vehicle k may
travel on a non-emergency delivery route.
xi) y., i = 1,...,N is the expected number of emergency deliveries
to hospital H. per period, y. is the probability that the
demand at H. exceeds the supply at H. given the optimal inventory
level at H. is used.
l
xii) s(&, q) is the systems cost function of a region, where I is the
number of hospitals in that region, and q is the amount of blood
used pejr year in that region.

126
William P. Pierskalla
xiii) y.., i = 1,...,N; j = N+1,...,N+M is a zero-one variable such that
y.. is 1 if hospital H. is assigned to central bank H. and is
0 otherwise.
xiv) x..,, i = 1,...,N+M; j = 1,...,N+M; k=l,...,n is a zero-one
i JK
variable such that x.., is 1 if vehicle k goes from hospital
H. to H. and is 0 otherwise.
The BTAP is:
PROBLEM 1:
N+M N+M n N N+M
min z (x,y) = E E E d. .x. .. + E E y.d. .y. .
1=1 j = l k=l J J i=l j=N+l J J
N+M N N
+ E s( E y. ., E Q.y..) (1)
j=N+l i=l ^ i = l VlJ
subject to
n N+M
E E
k=l j=l
N+M N
E E
j=l i=l
E E x.. = 1 i = 1, ... N (2)
ljk v '
Z Z QiXijk - Ck k = 1.... ,n (3)
N+M N+M
E E
j=l i=l
Z Z dijXijk±Dk k= 1 •" (4)
E E x. .. > 1 for all (S, S) (5)
ljk —
(i:H. eS} {j:H. eS} k=l
M containing jV
where S is any proper subset of «>» containing fr and S is the complement of
S.
N+M N+M
E x, ., = E x... k = l,...,n; h = 1,...N+M (6)
j=l h^k i=l lhk

SOME MATHEMATICAL MODELS IN HEALTH PLANNING
127
N+M N+M
Xii - E Xihk + E Xihk " l i = 1,..-,N; j = N+1.....N+M
1J h=l lhk h=l Jhk
k = l,..,,n
(7)
x. ., = 0,1
ljk
(note that x. .. =0)
nk
i = 1,...,N+M; j = 1,...,N+M;
k = 1,.. . ,n
(8)
y.. = 0,1
i = 1,...,N; j = N+l,.. .,N+M.
(9)
The explanation of these constraint sets are as follows. Constraints
(2) require that every hospital receive a shipment from some vehicle; (3)
are the vehicle capacity constraints; (4) are the maximum travel distance
constraints (note, it is implicitly assumed that Q. £ C, for i=l,...,N and
k=l,...,n); (5) require that graph ^C corresponding to x is connected; (6)
imply that a vehicle departs from a point h if and only if it enters there
(conservation of flow); (7) contains the coupling constraints between
variables x = {x..,} and y = {y..}. It means that if there is vehicle k passing
N+M
N+M
from both hospital i, (I x.,,= 1), and from bank j, ( E Xi,,
h=l lhk h=l Jhk
hospital i is assigned to bank j, (y
ij
1), then
_> 1 + 1-1 = 1). As shown in Or [1976],
these constraints imply that there is an optimal solution in which each
vehicle is based at a particular supply point.
In Problem 1 the variables x = {x..,} correspond to the routing of the
periodic delivery vehicles and the variables y = {y..} correspond to the
allocations of the hospitals to the blood banks. For a given x = l'x..,},
y = {y..} is uniquely determined, but the converse is not true; if we are
given the allocations, a series of M vehicle dispatch problems have to be
solved, in order to obtain the routings. Problem 1 has a finite feasible
solution set and a nonempty optimal solution set. However, the underlying
Multiple Vehicle Dispatch Problem (MVDP) makes it a complex integer
programming problem. . For N of any significant size (N >_ 20), the BTAP is too large
to be solved by conventional mathematical programming techniques in a
reasonable amount of time.
In Problem I, if y., i = 1,...,N are small or emergency costs
negligible (actual y.'s range from .0002 to .06 when optimal ordering policies are
followed, see Pierskalla and Yen [35]) and the function s(&, k) is essentially
constant, then

128
William P. Pierskalla
2 N+M N+M n
z (x) = E E E d. .x. ..
i=i j-i k=i ^ ^k
would be the dominating term in the objective function (1). Then we could
just solve the MVDP,
PROBLEM 2
N+M N+M n
min E Z E d. .x. ., (11)
i=l j=l k=l J J
subject to
n N+M
E E x = 1 i=l,...,N (12)
k=l j=l 1Jk
N+M N
E E
j=l i=l
E E Q.x. .. < C. k=l,. . .,n (13)
xi ijk — k
N+M N+M
E E d. .x. .. < D, k=l,. . . ,n (14)
j = l i=l ^ ^k~ k
n
{i:H.eS}{j:H.eS} k=l
E x. .. > 1 for all (S, S) (15)
ijk —
N+M N+M
E x, .. = E
3=1 1=1
E x, ., = E x.ul k=l,.. .,n; h=l,...,N+M (16)
hjk lhk v
x..k = 0, 1 i=l,...,N+M; j=l,...,N+M (17)
k=l,...,n
in order to obtain the optimal x for Problem 1. The optimal allocations,
• *
y , would then be uniquely determined by x .
On the other hand, if y., i=l,...,N are relatively large (which might
happen under nonoptimal ordering policies) or system costs and periodic

SOME MATHEMATICAL MODELS IN HEALTH PLANNING
129
delivery costs are negligible, then
N N+M
z (y) = E E y.d..y. .
1=1 j=N+l J J
would be the dominating term in the objective function. Then we could just
solve the allocation problem,
PROBLEM 3
N N+M
min I I y.d..y.. (18)
i-1 j=N+l X *> V
subject to
N+M
I y.. = 1 i = 1,...,N (19)
j=N+l 1J
y. . = 0, 1 i = 1,... ,N; (20)
1J j = N+l,... ,N+M
in order to get the optimal y° for Problem 1. Then, optimal routings, x°,
would be obtained by solving a vehicle dispatch problem for each one of the
M regions determined by y°.
* *
Let x be an optimal solution of Problem 2. Let y be the allocations
*
determined by x . Let y° be an optimal solution of Problem 3.
It directly follows from the above definitions that
zZ(x ) < zZ(x°)
z3(y°) < z3(y*)
and if the systems costs are essentially constant, then
z2(x*) + z3(y°)
would be a good lower bound on the optimal value of Problem 1.
In order to minimize the cost of operation under different regional
system configurations, the economies of scale curves representing feasible

130
William P. Pierskalla
combinations of the functional areas of blood banking were incorporated into
the model. These curves form the basis for determining the operating costs
for each of the different regional structures.
For example, if one considers the regional structure given by the first
figure in Figure 1, then the only questions which arise relative to costs
involve where should the community blood center be located in order to
minimize total transportation costs for donors, recruiting, phlebotomy on
mobiles, and routine and emergency deliveries to the transfusion services.
Since the other costs, such as processing, administration, inventory control
and phelbotomy at the Center are relatively independent of location, these
costs would not be included in the decision process, because they would be
incurred no matter where the community blood center was located.
On the other hand, if the regional structure is that of the second
figure in Figure 1, then all of the costs from the economies of scale curves
are relevant, since not only the location of the community blood centers but
also their sizes in the region are important decision variables.
Consequently, all of the curves are needed, and tradeoffs among these costs and
locations in sizes must be made.
Finally, if one were to analyze the third figure of Figure 1, the
appropriate use of the economies of scale curves would depend upon the authority
structure and governance relationships between the regional blood center and
the community blood centers. For example, if the regional blood center were
only a coordinating body of information and did not really have any authority
over the community blood centers, then the costs of operation would be very
similar to those in the preceding paragraph. That is, all of the functional
areas of blood banking would be located at the community blood centers;
consequently, virtually all of the costs would be incurred there, and hence
if one were to do a regional design, one would be interested in the location
of the blood centers, as well as their operating size. On the other hand,
if for example, donor recruiting were done at the regional blood center, then
the costs of donor recruiting would not be charged to the community blood
centers but would be a regional blood center cost. Since donor recruiting
costs are related to the distances the donor recruiters have to travel, then
the location of the regional blood center would be a factor in the cost
structure; however, none of the other costs, such as phlebotomy, processing,
inventory control and distribution administration at the community blood
Phlebotomy is the procedure of drawing blood from a donor. These drawings
often occur at the hospital, blood center, or at a distant location such as
a school, church, company, etc. When drawings are made at a distant
location, mobile blood vehicles are used and the drawings are thus called
"phlebotomy on mobiles."

SOME MATHEMATICAL MODELS IN HEALTH PLANNING
131
centers would be included in the cost of the regional blood center itself.
Consequently, wherever the functional areas and authority for administration
of those areas are located in the system, then those costs should be charged
at those appropriate places.
3. DECISIONS AT LOWER ECHELONS
The preceding analysis focused on macro and some middle level decision
making models in a regional system. In this section, we discuss decisions on
inventory control, issuing, and crossmatch release policies at lower levels
in the modeling hierarchy. These decisions impact the decisions at the
higher levels since they affect the system cost structures mentioned in the
previous section.
The basic community blood center scheme is that of a "wheel" structure
as shown in Figure 2, which is another view of Figure la.
Figure 2
The Community Blood Center and Its
Satellite Transfusion Service (TS's)

132
William P. Pierskalla
The community blood center acts as the hub of activities for its
satellite blood banks and transfusion services. In the centralized system,
the central blood bank supplies a number of TS's and maintains the authority
to redistribute all blood in the system. Most TS's maintain a supply of whole
blood and/or packed red cells. Other components may also be maintained at
some facilities.
Another way to view a CBB and the decisions needed to answer the
questions posed earlier is shown in Figure 3. In this figure, the CBB is shown
at the top and the lines represent the flow of units through the different
activities and TS's in the system. That is, Figure 3 is a schematic drawing
of the inputs, outputs, and flows in a centralized system. Some of the
decisions needed by the CBB are shown in the diamond-shaped boxes. The
variables S , S ,...,S represent the number of units needed on hand each day at
the CBB and at the TS's in order to meet the system needs without excess out-
dates. Of course, the S.'s are different for each ABO-Rh type and each
4 i
component and they change over time depending upon the changing needs at the
TS's.
Basically the inventory flow system for the CBB operates in the
following manner. Forecasts of future ABO-Rh blood needs and component needs are
made. The CBB periodically constructs mobile phlebotomy schedules and
forecasts the corresponding quantities to be drawn at each mobile site.
Individual drawings are also often made at the CBB itself and/or its TS's. These
drawings are scheduled to meet the forecasted demands at the TS's and maintain
a stock of inventories on hand at the CBB. On a daily, semi-weekly or weekly
basis depending on the level of activity and proximity to the CBB for each TS,
orders to the TS's must be filled.
After the CBB receives all the requests from the TS's, the orders are
filled by drawing from the inventories in the CBB. The decision policy as to
which units to send is called the issuing policy. The most common issuing
policies are first-in-first out (FIFO) or last-in-first-out (LIFO). For
purposes of simplification as well as good medical practice, each ABO type
and Rh factor is considered independent of the other types and Rh factors.
When the sum of all TS demands for whole blood/packed red cells (WB/PRC's) or
for particular components exceeds the total inventory in the CBB, the CBB may
backlog the excess demand or may fill all demands by calling in donors, by
contacting other CBB's, by using frozen packed red cells if appropriate, or
There are many derivative products which comprise whole blood and which can
be separated from the whole blood. These derivatives are called components
Some commonly derived components are: platelets, granulocytes, leukocyles,
cryoprecipitate, plasma and red cells.

SOME MATHEMATICAL MODELS IN HEALTH PLANNING
133
Other Central
or Regional
Blood Bank
FIGURE 3.
FLOW CHART FOR A CENTRAL BLOOD BANKING SYSTEM
SHOWING INVENTORY LOCATIONS AND CBB DECISION POLICIES

134
William P. Pierskalla
by requesting an emergency shipment from still higher echelon blood banks.
The CBB uses different approaches to handle the excess demand depending upon
whether the orders are routine or emergency.
Two examples of mathematical models to analyze the inventory levels for
fresh red cells, frozen red cells and platelets are given here. Models for
issuing and allocation of blood units are given in Yen [1975], Pierskalla and
Roach [1972] and Prastacos, et al., [1977].
4. A TWO PRODUCT PERISHABLE/NON PERISHABLE INVENTORY PROBLEM
As units of blood are drawn from donors they may be refrigerated, in
which case the shelf life is twenty-one days, or frozen, in which case the
shelf life is 365 days, which for all practical purposes is non-perishable.
When demands exceed the available supply of fresh blood, frozen blood may
then be thawed.
The One Period Model
We will make the following assumptions:
1) All orders are placed at the start of a period and received
instantly.
2) All stock arrives new.
3) Demands in successive periods are independent identically
distributed random variables with distribution F and density
f. In addition we will assume that f(t) > 0 when t > 0.
4) Inventory of product 1 (the perishable product) is
depleted according to a FIFO policy.
5) All costs are linear. They include:
a) Ordering in both products (at unit costs c and
c ) charged at the start of the period;
b) Holding in both products (at unit costs h and
h ) charged on what is on hand at the end of
the period;
c) Shortage (at unit cost r) charged on the
unsatisfied demand at the end of the period (the
shortage cost is either the cost of emergency
shipments from some source on the marginal
cost of an additional day in the hospital for
postponed surgery);
d) Outdating (at unit cost 0) charged on what
deteriorates at the end of the period.
6) If product 1 has not been depleted by demand before reaching

SOME MATHEMATICAL MODELS IN HEALTH PLANNING
135
age m-periods, it must be discarded at the unit cost
given in 5(d).
7) There is a single demand source. Demands first deplete from
product 1 (the perishable product) and then product 2.
Excess demand is backlogged in the second product.
A number of the assumptions may be relaxed or altered. It should be
noted that assumption (4) is quite mild. The optimality of FIFO for
depletion of perishable inventory has been established under far more general
circumstances (Nahmias [1974] and Pierskalla and Roach [1972]).
The approach will be to charge the outdating cost against the expected
outdating of the present order which will not occur for m periods. The
motivation behind this method is discussed in Nahmias [1975]. If
x = (x -,...,x) is the vector of perishable inventory on hand, x. = number
of units on hand which will outdate in exactly i periods, and y is the amount
of new perishable inventory ordered, then it has been shown that the
y
expression / G (u; x)du represents the expected outdating of y, m periods
into the future, where
t
G (t; x(n-l)) = / G ,(x . + v; x(n-2))f(t-v)dv
nv ~v JJ n-1 n-1 ~ v J
fl if t >_ 0
For each
0 if t < 0
n >_ 1, G (t; x(n-l)) is a C.D.F. in its first argument, and may possess a
discontinuity at t = 0.
2
Letting x = amount of product 2 on hand,
z = amount of product 2 on hand after ordering,
the total expected cost of ordering y of product 1 and z - x of product 2
x+y y
c y + h / (x+y-t)f(t)dt + 6 / G (u; x)du + c (z-x ) + h zF(x+y)
1 0 0 m ~
I
h2 / (x+y+z-t)f(t)dt + r / (t-x-y-z)f(t)dt
x+y+z
/
x+y x+y+z
m-1
2
where x = Z x. for convenience. Collecting all terms independent of x we
i=l 1

136
William P. Pierskalla
2
will write this as L(x, y, z) - ex . A point which should be noted here is
that the decision variables for each product have different interpretations:
y represents the actual quantity of product 1 ordered, while z is the inventory
level of product 2 after ordering. Our interest in the single period model is
secondary to that of the multi-decision dynamic problem. The optimal ordering
policy over the finite horizon will satisfy the functional equations
oo
C (x, x2) = inf U(x, y, z) - c?x2 + a/C - (s (x, y, t), s (x+y, z, t))f(t)dt}
n ~ y^Q l Q n-i ~i ~ i.
, 2
z>x
for 1 <_ n <_ N(N is a fixed positive integer). The transfer functions are
given by:
s (x, y, t) = (s (x, y, t),...,s (x, y, t))
~1 ~ 1,m-l ~ 1,1
where
Sl
. (x, y, t) = (x - (t - I x ) + )\ 1 <_ i <. m-1
' ~ 1+i j=1 J
(we interpret x = y)
and s^(x+y, z, t) = z - (t-(x+y)) where g = max(g, 0). The discount factor
is a c(0, 1).
2
Again collecting all terms independent of x , we will write
2 ->
C (x, x") = inf IB (x, y, z) - ex")
y>_0
z>x2
in which
BR(x, y, z) = L(x, y, z) + a f°° C x(s (x, y, t), s2(x+y, z, t))f(t)dt.
0
2
The functions C (x, x ) have the usual interpretation as the minimum expected
discounted cost for an n period problem when (x, x ) is on hand. As is
customary with dynamic programming models, the periods are numbered backwards.
The goal of our analysis will be to answer the two questions: when should
an order be placed and how much of each product should be ordered?

SOME MATHEMATICAL MODELS IN HEALTH PLANNING
137
The Ordering Regions
It becomes notationally and mathematically convenient to introduce the
assumption that inventory remaining at the end of the horizon can be salvaged;
inventory of product 1 remaining may be salvaged at a return c.x and of pro-
2
duct 2 at a return c?x . Backlogged demand in product 2 may be made up by an
2
emergency order at a cost - q x . That is,
CQ(X, X ) = - CXX - C2X .
The assumption is identical to that made by Veinott [1965] in the analysis of
non-perishable multi-product problems. The convenience of this assumption
is that the somewhat artificial fact that the problem terminates in a given
time horizon with inventory remaining is removed and the problem behaves more
like a problem under steady state conditions. In some inventory problems this
assumption allows the decomposition of the n-period problem into n one-period
problems which are easier to solve.
In addition we will make the following four assumptions regarding the
cost parameters:
i) 0 £ h2 <_ h
ii) 0 < c < c2
iii) r > (l-a)c2
iv) 0 <^ (l-a)(c2 - cp + (h2 - hx) < 6
Since perishable inventory must often be stored under special conditions,
assumption (i) is not unreasonable. Assumption (ii) is necessary to insure
that it is economical to stock product 1. Assumption (iii) is a usual one
made for non-perishable inventory (Arrow, et al., [1958]). The expression
of assumption (iv) is precisely the difference in the costs of procuring and
holding one unit of each type of inventory and salvaging it the following
period. If this term were negative then it would never be optimal to order
in product 1 ; or if it exceeded the unit cost of outdating, it would never
be optimal to order to a positive level in product 2.
We will have need to refer to the following constants in the analysis
of the ordering regions in the first period:

138
William P. Pierskalla
-F-1
p-Cj(l-a) + (h2 - hjfl
When F is strictly increasing, assumptions (iii) and (iv) guarantee that
* *
u and w both exist and are strictly positive. If the additional condition
ac? - h? < c is satisfied (although we will not require it to be), then we
may also define the constant
F"1
|~r - Cl * aC2"|
With the inclusion of the salvage value assumption it follows from the
definition of the transfer functions that
m-1
B (x, y, z) = L(x, y, z) + aF(x )[- c (y + Z x.) - c z]
i ~ ~ i i i=2 i z
x+y
- acx / (x+y-t)f(t)dt - ac2l>2(x+y) - Ffx^Jz
Xl
CO
- ac2 / (x+y+z-t)f(t)dt)
x+y n 1
We will adopt the following notational convention: if h: R -> R and
he C^ , then h^1' is the first partial derivative of h with respect to its
ith argument and hr '-^ is the second cross partial derivative with respect
.th , .th •
to the i— and j— arguments respectively.
We have the following
THEOREM II.4.1: B (x, y, z) is convex in (y, z) for all non-negative x. The
functions y (x), z_ (x) solving B. (x, y (x), Z-fx))= miniB.(x, y, z)} satisfy
, •. 1~ 1 ~ r i ^ v, z 1 ~
Bl ^X> yl ^ * zi^x^ = Bi (x> Xi M > zi(x)) = 0'and are unique for all x.
Since y (x) > 0 for all x, a necessary and sufficient condition that it
1 ~ 2
be possible to order to the global minimum of B (x, y, z) is x < z (x). If
2 ■*• ~ 1 ~
x t. z (x), it may still be optimal to order product 1. In this case we
1 ~ 2
define the function p (x, x ) to satisfy
2 2 2
Bx(x, px(x, x ), x ) = inf(Bx(x, y, x )).
y
2
If p1(x, x ) > 0 then it is optimal to order this amount of product 1.

SOME MATHEMATICAL MODELS IN HEALTH PLANNING
139
Hence the following characterization follows:
THEOREM II.4.2: A necessary and sufficient condition that it is optimal to
order a positive amount of product 1 is B: (x, 0, x ) < 0. If one reasons
analoguously to THEOREM II.4.2, then it is tempting to assume that
BJ (x, 0, x ) < 0 implies it is optimal to order in product 2. However
this is not the case. To see why, let t(x, y) satisfy B|m+ ^ 0,y,t(x,y)) = 0.
Differentiating implicitly with respect to y we obtain
fm1 - B.(m+1' m) (x, y, t(x, y))
tl J (x, y) = -~ ^— - < 0.
B(m+1, m+l) (x> y> t(;> y))
Since t(x, y,(x)) = z (x) and y (x) > 0 it follows that t(x, 0) > z (x). If
x satisfies z (x) <^ x < t(x, 0) then BJm J (x, 0, x ) < 0 and it is optimal
not to order in product 2.
Hence there are exactly three distinct ordering regions:
2
Region I - Optimal to order in both products: x < z (x),
2
Region II - Optimal to order in product 1 only: x >^ z (x)
Bjm)(x, 0, x2) <0(P1(x, x2) > 0.
Region III - Optimal not to order: B|m^(x,0,x ) >_ 0, (p (x, x ) <_ 0.
Note from the proof of THEOREM II. 4. 2 that if it is optimal to order in
product 2 then it is optimal to order in product 1. The boundary between
2
Regions I and II is quite complex, as it depends on the entire vector (x, x ).
However the boundary between Regions II and III depends on the vector x only
through the sum of its components, x.
Define g(x) = (l/r+h2)) • {r + ac2 - ^ - F(x)[(h1 - h2) + a(c2 - c^]).
Then
2
THEOREM II.4.3: A necessary and sufficient condition that (x, x ) is in
Regions I or II is that
F(x + x2) < g(x).
*
The function g(x) is strictly decreasing in x with g(+°°) = F (w ). If
*
the condition ac? - h? < c is satisfied then F(w ) < g(x) < 1 for all x >_ 0
and v = F (g(0)) will exist. However, if ac2 - h >^ c then g(0) >_ 1. In
this case there exists a unique number p >_ 0 which solves g(p ) = 1. The
2
boundary between Regions II and III may be pictured in the (x, x ) plane

140
William P. Pierskalla
*♦ x^F'^glx)]
REGION 111
REGION
-x* x* = u -y1 (x)
x ♦ x
2 = u*-y, (x)
[a] aC2-h2<Ct
[b] a C2 - h2^C,
FIGURE 4. THE ORDERING REGIONS

SOME MATHEMATICAL MODELS IN HEALTH PLANNING
141
independent of m. Figure 4 pictures this boundary for each of the two cases
2
above. The boundary between Regions I and II may be pictured in the (x, x )
plane only if m = 2. The arrows indicate the inventory position after order-
2 *
ing. Notice that (x, x ) e Region I guarantees that x + y (x) + z (x) = u
(which follows from B|m+ ' (x, y (x), z (x)) = 0) while (x, x ) £ Region II
will yield F(x + x + p,(x, x )) < g(x + p,(x, x )).
These results can be extended to the multiperiod dynamic problem.
Even with the inclusion of the salvage value assumption, neither the ordering
regions nor the ordering policies are stationary in time.
5. A BY-PRODUCT PRODUCTION SYSTEM WITH AN ALTERNATIVE
The idea behind this example is that components are made from whole
blood, e.g., red cells and platelets. In such cases, two simultaneous
decisions must be made. First, the inventory level of each product must be
determined to best meet (random) demands. Second, the best (optimal)
production level (e.g., number of runs) must be determined to achieve these
inventories. Often these decisions cannot be made independently because
production capacities or design characteristics of the production lines may
prohibit arbitrary combinations. The system we consider is describe in
Figure 5. There are two inventory items, called products 1 and 2, and two
different production processes called types A and B. Type A is capable of
making both products, simultaneously, according to the production coefficients
r, and r\ . When type A is operated at the unit level (e.g., a single run),
n, and T\ units of products 1 and 2 are obtained. We choose to call type A
a by-product production process due to its multi-product capability. Type B
is a single item production process and is capable of making only product 2.
In this context, type B is the alternative method of obtaining product 2.
PRODUCT 1
V
STOCHASTIC 1
DEMAND 1
Xn2
TYPE B
PRODUCT 2
fr
STOCHASTIC
DEMAND 2
FIGURE 5
Block Diagram of the Production System

142
William P. Pierskalla
The model is of the periodic review type, where the planning horizon is
N periods long. At the beginning of period n, n=l,2,...,N, the initial
inventories (before production) x = (x ,. x „) are reviewed, where x .is
r n n,l n,2' n,j
the initial inventory of product j, j=l»2. Then, the starting inventories
(stock levels after production but before demand) y = (y ., y 2) and the
the production levels p = (p ., .p _) are jointly determined, y . is the
r *n vrn,A* *n,lr J J ;nj
starting inventory of product j, j = l,2 in period n and p , p the
production levels of types A and B, respectively. Then, a random demand
D = (D , D ) is realized. We use a reverse recursion numbering scheme so
that index N refers to the first period and 1 refers to the last period. When
focusing attention to quantities within a period, we frequently drop the
period index on the above quantities.
The following is a list of specific assumptions governing the development
of our model.
1) Production Processes
It is assumed that each production process is controlled by specifying
the production level. For type A this level is p • for type B it is pR. Thus,
the amount of each product produced is (n,pA, PR + ^?PA). We assume that r\
and ti are constant; they are not decision variables. For the present, we
assume that there are no lead times and that there are no upper bounds on
production.
2) Demands
We assume that D , n=l,2,...,N form a sequence of nonnegative independent
and identically distributed random vectors with a continuous joint density
function f(-) and finite expected value. Although we allow dependencies among
demands within a period, we do require that D . be satisfied only from
inventories of product j, j=l,2. It is further assumed that unsatisfied
demands are backlogged and are not lost.
3) Production Costs
a) c. is the cost per unit of production on A;
b) c is the cost per unit of production on B;
D
c) there are no fixed costs of production.
4) Inventory Costs
Let g.(-) be the holding and shortage cost for product i, i=l,2, over
any single period. We define the expected total holding and shortage over any
period as given by

SOME MATHEMATICAL MODELS IN HEALTH PLANNING
143
L(y) = / / [g (y - t) + g (y - u)] f(t, u)dt du.
0 0 l l Z Z
We assume that
a) L(-) is strictly convex over R '
b) L(-) has continuous second partial derivatives;
C) ffj- (CA * Vb^I + CBy2 + LM "+°°
whenever ||y|| -* +00 (where ||»|| is the Euclidean norm). (3) assures the
existence of an optimal policy.
5) Discount Parameter
Let a e [0, 1]. a denotes the parameter which relates costs in future
periods to the present.
We use the following notation for derivatives and partial derivatives of
functions. For a twice continuously differentiable scalar function h(x) , let
h'O) and hn(-) represent its first and second derivatives, respectively. For
a function g(«) defined on R with continuous second partial derivatives, let
gU) (x) = 3~- g(x) , i=l,2,...,n
i
g(l,3)(x) = 8x3 9x g(x)' i,j=l,2,...,n.
J i
The problem is to determine a policy that leads to the minimum expected
discounted costs over the N period horizon. This leads to an optimization
problem that is most easily conceptualized as a dynamic programming problem.
Before formulating the dynamic program, we first show the relationship between
a feasible set of production levels p = (p., pR) and starting inventory levels
Y - (vi> y?) given any initial inventory x. Given p and x, the starting
inventories y must satisfy
yx = VA + xi
y2 - Va + pB + v
Given y and x, the production levels p are determined by

144
William P. Pierskalla
PA - f^ ^1 " V
PB = ^2 - nf V " (X2 '^ V = y2 - (X2 + ^ &l ' Xl»
Although the dynamic program is most easily defined in terms of p anx x, we
favor using y and x since they characterize the optimal policy.
The dynamic programming recursion is given by
cn(x) = inf {Gn(y) - J- (cA - r^c^ - c^}
subject to
yi ±xi
n2
for
y21 x2 ♦ — (Xl - Xl)
GnM = f^ (CA " n2CB)yl + cBy2 + L(y)
+ a / / C (y - t, y - u)f(t, u)dt du
0 0 z
where n=l,2,...,N, y e R and CLfx) = 0 for all x.
The first result is of a purely technical nature, but provides the
properties of C (•) and G (•) crucial to the characterization theorem.
THEOREM II.5.1: Under the assumptions made above, the following hold.
1. G (•) is continuous and strictly convex.
2. All sets of the form Q (3) = (y e R2,# G (y) _< 3} are
compact and convex for each bounded 3 £ R.
3. The point y (x) that solves the constrained dynamic program
exists and is unique. Hence, inf can be replaced by min for
each bounded x e R? and n=l,2,...,N.

SOME MATHEMATICAL MODELS IN HEALTH PLANNING
145
4. C (•) is convex and nonneeative on R~.
nv J & 2
The optimal policy, y (x) is completely characterized by a point
S = (S ,, S 2) and three scalar functions r (•), w (•), and q (*) as
graphically described in Figure 6. Each arrow represents the path from an
initial inventory (the tail) to its beginning inventory (the head) by some
production combination. In Region I both A and B are used and notice that the
starting inventory level will always be S . In Region III only type B is used
so the starting stock always lies on the graph of q (•). In Region II only A
is used. The path from x to y (x) in this region is a line parallel to w (•).
It is also important to point out that r (•), w (♦) and q (•) are functions of
v
Many of the assumptions in this example can be relaxed or removed so the
model is more realistic. The demands and costs may be nonstationary, the
production capacities may be finite, unmet demands may be lost, fixed lead
times may be incorporated and the general m-process-n-product case follows
naturally.
In summary, then the analysis presented in these section allows one to
look at the economics of regionalization, including the location of community
blood centers, the allocation of hospital blood banks and transfusion services
to them, and the costs of different structures for a region. As mentioned at
the beginning, these are not the only variables which one would want to
consider in implementing a regional system. However, they are extremely
important variables since they affect the system operation and its short and
long-run costs. These costs in turn affect the subsequent charging mechanisms
for blood and components to the patient. This quantitative analysis is a
guide to making important decisions in a more enlightened manner.
6) Concluding Remarks
Finally, it is hoped that the preceding analysis has communicated to
the reader some of the flavor of past and current research in several
pertinent methodological areas of Operations Research: Inventory Theory,
Facility Location, Vehicle Scheduling, and to a small extent Mathematical
Programming and Optimization of Stochastic Models. In addition it is hoped
the reader will be interested in pursuing the design and analysis of models
which better measure health care delivery phenomena and which lead to better
decisions for individuals and institutions in this important societal area.

146
William P. Plerskalla
H
Q
§
P*
Sn, 2
REGION
II
A ONLY
„r (x)
n
REGION IV
NO PRODUCTION
i<*W
REGION
I
A & B
Sn, 1
REGION
III
B ONLY
q (x)
Product I
FIGURE 6
Characterization of the Optimal Policy in Period n

SOME MATHEMATICAL MODELS IN HEALTH PLANNING
147
REFERENCES
Arrow, K.J., S. Karlin and H. Scarf, eds. Studies in the Mathematical Theory
of Inventory and Production, Stanford University Press, Stanford,
Calif. (1958).
Barlow, R., L. Hunter and F. Proschan, "Optimum Checking Procedures," J. Soc.
Ind. Appl. Math., 11, 4 (1963), pp. 1078-1095.
Calvo, A.B. and D.H. Marks, "Location of Health Care Facilities: An
Analytical Approach," Socioeconomic Planning Science, 7 (1973), pp. 407-
422.
Cohen, M.A. and W.P. Pierskalla, "Managment Policies for a Regional Blood
Bank," Transfusion 15 (1975), pp. 58-67.
Cohen, M.A. and W.P. Pierskalla, H.S. Yen, I. Or, "Regionalization and
Regional Management Strategies in the Blood Banking System: An Overview."
Proceedings of the Second National Symposium on the Logistics of Blood
Transfusion Therapy, June 19-20, 1975. Sponsored by the Michigan
Association of Blood Banks and the Michigan Community Blood Center.
Cooper, L., "Location-Allocation Problem," Operations Research, 11 (1963).
Cooper, L. , "Hueristic Methods for Location-Allocation Problems," SIAM Review,
6 (1964).
Cooper, L., "The Transportation-Location Problem," Operations Research, 20,
(1972).
Derman, C., "On Minimax Surveillance Schedules," Naval Research Logistics
Quarterly, 8 (1961), pp. 415-419.
Deuermeyer, B. and W.P. Pierskalla, "A By-Product Production System With An
Alternative," Management Science, 24, (1978), pp. 1373-1383.
Eddy, David M. , "Screening for Cancer: Theory, Analysis, and Design,"
Department of Family, Community, and Preventive Medicine, Stanford
University, (1978).
Eddy, David M., "Implementation Plan: Part III. Rationale for the Cancer
Screening Benefit Program Screening Dolicies," Department of Family,
Community, and Preventive Medicine, Stanford University, for Blue Cross
Association, (1978).
Francis, R.L., and J.A. White, Facility Layout and Location, Prentice-Hall,
Inc., Englewood Cliffs, N.J. (1974).
Fries, B., "Optimal Ordering Policy for a Perishable Commodity With Fixed
Lifetime," Operations Research, 23 (1975), pp. 46-61.
Hutchinson, G. and S. Shapiro, "Lead Time Gained by Diagnostic Screening for
Breast Cancer," Journal of the National Cancer Institute, 41 (1968), p.
665.
Jennings, J., "Blood Bank Inventory Control," Management Science, 19 (1973),
pp. 637-645. Also appears in Analysis of Public Systems, R. Keeney,
P. Morse (editors), MIT Press, Cambridge, Mass., (1972).

148
William P. Pierskalla
Jennings, J., "Inventory Control in Regional Blood Banking Systems," Technical
Report Number 53, Operations Research Center, MIT (1970).
Keller, J., "Optimum Checking Schedules for Systems Subject to Random
Failure," Management Science, 21 (1974), pp. 256-260.
Kirch, R. and M. Klein, "Surveillance Schedules for Medical Examinations,"
Management Science, 20 (1974), pp. 1403-1409.
Kirch, R. and M. Klein, "Examination Schedules for Breast Cancer," Cancer,
33 (1974) , pp. 1444-1450.
Kolesar, P., "A Markovian Model for Hospital Admission Scheduling," Management
Science, 16, 6 (February 1970), pp. B-384-396.
Lincoln, T., and G.H. Weiss, "A Statistical Evaluation of Recurrent Medical
Examinations," Operations Research, 12 (1964), pp. 187-205.
McCall, J., "Maintenance Policies for Stochastically Failing Equipment: A
Survey," Management Science, 11 (1965), pp. 493-524.
McCall, J., "Preventive Medicine Policies," Rand Corporation, P-3368-1,
(1969).
Magnanti, T., B.L. Golden, and H.Q. Nguyen, "Implementing Vehicle Routing
Algorithms," Technical Report No. 115, Operations Research Center,
Massachusetts Institute of Technology (September 1975).
Miller, H.E., W.P. Pierskalla, and G.J. Rath, "Nurse Scheduling Using
Mathematical Programming," Operations Research, 24, 5 (September-October
1976), pp. 857-870.
Nahmias, S. and W.P. Pierskalla, "Optimal Ordering Policies for a Product
That Perishes in Two Periods Subject to Stochastic Demand," Naval
Research Logistics Quarterly, 20 (1973), pp. 207-229.
Nahmias, S., "Optimal Ordering Policies for Perishable Inventory,"
Operations Research, 23 (1975), pp. 735-749.
Nahmais, S., "Inventory Depletion Management When the Field Life is Random,"
Management Science, 20 (1974), pp. 1276-1283.
Nahmais, S. and W.P. Pierskalla, "A Two-Product Perishable/Nonperishable
Inventory Problem," SIAM J. Applied Math., 30, No. 3 (1976), pp. 483-500.
Or, I., "Traveling Salesman-Type Combinatorial Problems and Their Relation
to the Logistics of Regional Blood Banking." Doctoral Dissertation,
Northwestern University, Evanston, Illinois - 60201 (1976).
Or, I., and W.P. Pierskalla, "BTAP: A Computer Program to Obtain Solutions
to the BlQod Transportation-Allocation Problem and Other Traveling
Salesman Type Problems," Technical Report, Department of Industrial
Engineering and Management Sciences, Northwestern University, Evanston,
Illinois (August 1976).
Or, I., and W.P. Pierskalla, "A Transportation Location-Allocation Model for
Regional Blood Banking," AIIE Transactions, (Summer 1979).
Pierskalla, W.P. and J.A. Voelker, "A Survey of Maintenance Models: The
Control and Surveillance of Deteriorating Systems," Naval Research
Logistics Quarterly, 23, 3 (1976), pp. 353-388.

SOME MATHEMATICAL MODELS IN HEALTH PLANNING
149
Pierskalla, W.A. and J.A. Voelker, "A Model for Optimal Mass Screening and the
Case of Perfect Test Reliability," Technical Report No. 3, Department of
Industrial Engineering and Management Science, Northwestern University,
Evanston, Illinois - 60201 (1977).
Pierskalla, W.P. and C. Roach, "Optimal Issuing Policies for Perishable
Inventory," Management Science, 18 (1972) , pp. 603-615,
Pierskalla, W.P. and J.A. Voelker, "Mass Screening With a Heterogeneous
Population," Working Paper, Department of Industrial Engineering and
Management Sciences, Northwestern University, Evanston, Illinois (1978).
Submitted to Operations Research.
Prastacos, G.P. and E. Brodheim, "A Regional Blood Distribution Model," Working
Paper, The Wharton School, University of Pennsylvania, Philadelphia,
Pennsylvania (1977).
Prastacos, G.P., "Allocation of Perishable Inventory," Technical Report No. 70,
Operations Research Group, Columbia University, New York, New York
(January 1977).
Prorok, P., "On the Theory of Periodic Screening for the Early Detection of
Disease," Unpublished Ph.D. Dissertation, University of New York at
Buffalo (1973).
Prorok, P., "The Theory of Periodic Screening, I: Lead Time and Proportion
Detected," Advances in Applied Probability, 8 (March 1976), pp. 127-143.
Prorok, P., "The Theory of Periodic Screening II: Doubly Bounded Recurrence
Times and Mean Lead Time and Detection Probability Estimation," Advances
in Applied Probability, 8 (September 1976), pp. 460-476.
Roeloffs, R., "Minimax Surveillance Schedules With Partial Information,"
Naval Research Logistics Quarterly, 10 (1963), pp. 307-322.
Roeloffs, R., "Minimax Surveillance Schedules for Replacement Units, Naval
Research Logistics Quarterly, 14 (1967), pp. 461-471.
Schwartz, M. and H. Galliher, "Analysis of Serial Screening in an Asymptomatic
Individual to Detect Breast Cancer," Technical Report, Department of
Industrial and Operations Engineering, College of Engineering, University
of Michigan (1975).
Thompson, D. and R. Disney, "A Mathematical Model of Progressive Diseases and
Screening," presented at the November 1976 ORSA/TIMS Meeting.
Vienott, A.F., Jr., "Optimal Policy for a Multi-Product, Dynamic, Non-
stationary Inventory Problem," Management Science, 12 (1965) , pp. 206-222.
Vienott, A.F., Jr., "The Status of Mathematical Inventory Theory," Management
Science, 12 (1966), pp. A745-A777.
Voelker, J.A., "Contributions to the Theory of Mass Screening," Ph.D.
Dissertation, Northwestern University, 1976.
Wendell, R.E. and A.P. Hurter, "Location Theory, Dominance, and Convexity,"
Operations Research, 21 (1973), pp. 314-320.
Wendell, R.E. and A.P. Hurter, "Optimal Locations on a Network," Transportation
Science. 7 (1973), pp. 18-33.

150
William P. Pierskalla
Yen, H., "Inventory Management for a Perishable Product Multi-Echelon System,"
Ph.D. Dissertation, Northwestern University, 1975.
Zelen, M. and M. Feinleib, "On the Theory of Screening for Chronic Diseases,"
Biometrika, 56 (1969), p. 601.
Zelen, M., "Problems in the Early Detection of Disease and the Finding of
Fault," presented at the meeting of the International Statistics Institute,
Washington, D.C. (1971).

Proceedings of Symposia in Applied Mathematics
Volume 25, 1981
OPERATIONS RESEARCH: APPLICATIONS IN AGRICULTURE
Robert B. Rovinsky
U.S. Department of Agriculture
Christine Shoemaker
Cornell University
ABSTRACT: Operations Research techniques have been applied
to a wide range of problems arising in agriculture. After
surveying applications to a variety of agricultural problems,
we focus on a review of mathematical models used for
pesticide policy analysis.
In the following discussions we will distinguish between two types of
agricultural problems, which differ both in the role mathematical researchers
can take and in the methodologies they may employ. The first are "farm level"
problems, a term which has traditionally been used to describe decisions made
at the local, basic unit of agriculture-the farm. However, the definition
may be broadened to include decisions by groups, including individual
agribusiness firms, farm cooperatives, and forest managers. In these instances
the "goals" of the decisionmaker are usually clear and can often be
mathematically specified, as can the constraints and conditions under which the system
works. The form of analysis will depend upon such things as the level of
uncertainty, the boundary conditions, and the data available. The second type
of agricultural problem concerns "policy." In such problems there may be
several competing decisionmakers, each with different goals. In addition, it
may not be possible to precisely specify the constraints and other
relationships among variables.
Table 1 lists a few of the applications of operations research in
agriculture. Day and Sparling (1977), Johnson and Rausser (1977), and Rausser
et al. (1980) give a more detailed review. Their articles contain selected
bibliographies totalling over 500 published applications of Operations Research
in Agricultural Economics.
In what follows, we will focus on mathematical models and Operations
Research techniques used in pest management, both at the local level where
insect and weed damage must be controlled by a systematic use of chemical
1980 Mathematics Subject Classification 90B99
Copyright © 1981 American Mathematical Society
151

152
R.B. Rovinsky and C.A. Shoemaker
and/or biological controls, and at the regional or national level, where
economic and environmental policy issues must be examined using large-scale
models of entire sectors of the agricultural system. We will confine our
discussion to three recent studies in which we have been heavily involved. The
first concerns applications of optimization methods, especially dynamic
programming, to pest management. The second involves the use of quadratic
programming to study how cooperation among farmers can increase their profits
and reduce their risk under a variety of pest management policies, and the
third focusses on the evaluation and selection of policy models by the
Department of Agriculture in its attempt to respond to environmental
restrictions on pesticide use.

APPLICATIONS IN AGRICULTURE
153
Table 1: Examples of Applications
Farm Level Applications
Type of Problem
Farm Management
Diet Selection
Crop Rotation
Comparative Static Analysis
Farm Budgets, Economics of
Soil Conservation
Transportation
Farm Investment
Risk
Weather
Notes on Technique and Applications
-First formulated in 1941, solved by
Linear Programming (LP) in 1947. for
animals in 1951, now a routine service
available at Agriculture Experiment
Stations.
-One of the first applications of linear
programming.
-Parametric programming
-General mathematical programming,
including integer, nonlinear, mixed
integer.
-Many types of analyses including
network theory. International trade
flows are a current problem area.
-Dynamic programming, multiple optimal
replacement theory.
-Stochastic programming
-Game theory, probabilistic modeJling
(Also, computer-aided Real Time Management Systems are new aval lable at Purdue and
Michigan State Universities, and are used by Extension Service personnel.)
Farm Firm Analysis
Farm Growth Models, Effect
of Development Policies
Production Response
Regional Adjustment
Dynamic Allocation
Short-run national planning
-Recursive programming (a variation of
linear programming; a series of LP problems
are solved until a desired economic
equilibrium is achieved.)
Policy Applications
-Linear programming, cluster analysis,
dual linear programming.
-Recursive linear programming
-Multisector recursive linear
programming.

154
R.B. Rovinsky and C.A. Shoemaker
Interregional and Spatial Economics
Distribution and Pricing
Efficient allocation
Optimal Location of Processing
Plants
National Allocation of Resources
Spatial Equilibrium Prices
Market Demand and Prices in
Spatial Equilibrium
-Early use of Hitchcock-Koopmans
transportation model.
-LP and network theory
-Parametric and reactive programming
-Quadratic programming
Agricultural Development
Foreign exchange maximization
Large scale development
Small scale development
-Input/output analysis
-Mixed integer LP, decomposition LP,
simulation.
-Game theory, linear programming
Applications of Optimization Methods to Pest Management
Crop losses due to pest damage (primarily insects and weeds) have been
estimated by the U.S. Department of Agriculture to be 30 percent of production,
or about 55 billion dollars in 1974. This is in spite of the fact that almost
one billion pounds of pesticides, worth over one billion dollars, were used in
that year. The pest problem is even more serious in tropical countries, where
disease and malnutrition are directly related to insect and weed infestations.
The environmental effects of pesticides and the health hazards they pose to
farmworkers have been well documented. Several major pesticides, including
DDT, have been withdrawn from use in the United States for health or
environmental reasons.
In an attempt to reduce pesticide use and the associated environmental
hazards, agriculturalists have utilized a number of non-chemical pest control
methods. Cultural pest control methods like destruction of crop remains and
changes in harvest dates have been encouraged. The use of biological control
agents like insect predators or parasites have also been an important part of
pest control programs. The co-ordination of cultural biological, chemical
and other means of pest control is called integrated pest management.
Mathematical analysis has been used as part of interdisciplinary research
investigations to determine the best ways to integrate the timing and scope of
available pest control alternatives.
The dynamics of plant and pest growth and damage levels are complicated.
For example, the optimal timing of an insect pest management strategy strongly

APPLICATIONS IN AGRICULTURE
155
depends on the age distribution of the pest population, densities of beneficial
and pest population, condition of the crop, and temperature, among other
factors. One approach has been to develop mathematica1 simulation models,
which attempt to synthesize crop and pest behavior with mathematical equations
relating each component of the ecosystem. These models require estimation of
three related components—the growth or yield of the crop, the growth or decline
of the destructive pest population, and the control, whether it be pesticide
applications or the behavior of a prey or parasite species.
For several years, there has been a considerable interest in the use of
optimization methods to aid in the development of efficient pest management
programs. Given the complexity of the agricultural ecosystem and the
stochastic nature of important driving variables such as weather, the simulation
models which have been developed to describe the effect of management on the
dynamics of the pest and/or the crop typically have a very large number of
variables. Because of the cost of solving a simulation model for each of the
large number of combinations of control decisions that are possible in pest
management analysis, optimization methods have been used to aid in the
selection of control policies. The basic mathematical challenge associated with
the application of optimization methods to this problem is the development of
computationally feasible procedures for such large, complex systems.
Discussions of the applications of optimization methods to pest
management analysis have been included in review articles by Jaquette (1972) , Conway
(1977) , and Wickwire (1977) . These applications have utilized a variety of
optimization methods including a gradient and quadratization technique,
Powell's method, a reduced gradient method (Regev et al., 1977), and dynamic
programming (Shoemaker, 1973, 1977; Taylor and Headley, 1975). Dynamic
programming (Bellman and Dreyfus, 1962) is the technique which has been most
widely used because it is especially well suited for analysis of pest control
decisions which are made at discrete points in time in a stochastic,
observable environment. However, each of the optimization procedures have different
advantages and disadvantages. Hence the selection of an optimization method
depends upon the mathematical structure of the pest management model. For the
sake of brevity we will restrict our review below to several recent
applications of dynamic programming. These applications illustrate the procedures
used to circumvent the dimensionality problems associated with the
application of optimization methods to pest management.
Dynamic programming is an optimization technique which is based upon the
solution of recursive equations describing the decision process at each of a

156
R.B. Rovinsky and C.A. Shoemaker
number of stages. For a stochastic decision problem the form of the recursive
equation is:
Hk(sk) = Min[R(s\ vk) + a Z P[sk+1 = a).|s\ vk ] Hk+1 (ai. ) ] ] (1)
k i=1 -i i
v
k k k k
where s is the state vector in period k, v is the decision vector and H (s ) is
the minimum cost of going from period k to the end of the decision horizon.
k k
R(s , v ) is the net return (e_.£., profit minus cost) received in period k.
~" k+1 ~" k k k+1
P[s =aj-j|s > v ] is the probability that the state s will equal w. given
"" k k 1
that the state in period k is s and the control decision v . The constant a
is a discount factor. In the deterministic case, the equation can be more
simply written as:
Hk(sk) = Min[R(sk, vk) + a Hk+1(G(s\ vk))] (2)
th k+1 k k
where the i component of the state vector s. = G.(s , v ). The components
of the state vector s are usually used to describe the age structure of the
plant or pest population, as well as variations in spatial distributions or
genetic composition.
One pest which has been the object of several optimization studies is the
spruce budworm. Epidemic outbreaks of this pest, which occur approximately
every forty or fifty years, have resulted in tree mortality on hundreds of
thousands of acres in eastern Canada and the United States. Large simulation
models have been created to describe the effect of pest control practices on
the occurrence of outbreaks and on tree mortality (Jones, 1977; Stedinger,
1977). These models are very complex, involving several hundred geographical
locations, and over twenty variables at each site to describe budworm densities9
tree damage and tree age structure. The time steps of the models are one year,
and the model is computed for at least one hundred years to evaluate the impact
of alternative control strategies.
Attempts to utilize optimization methods for analyzing spruce budworm
management have required a reduction in the number of state variables over
those used in the simulation models. Using a method suggested by Dantzig (1974),
Winkler (1975) developed a dynamic programming model with two state variables.
The first variable s is the density of spruce budworm eggs at the start of the
th k
k year and s is a measure of the amount of defoliation of the trees which
had occurred in the previous year. The problem formulation is similar to that
given in Equation (1), where the probability P reflects the impact of
stochastic weather events on budworm reproductive and mortality rates. In
year 1 all trees are assumed to be one year old. The two most important

APPLICATIONS IN AGRICULTURE
157
factors which are ignored in this description are the mixed age structure of
the forest and the spatial distribution of budworm densities. As pointed out
by Holling and Dantzig (1978), this formulation does not incorporate the fact
that a pesticide application cannot be directed only at trees of a fixed age,
but rather it must be applied to an entire stand, in which tree ages usually
vary over a considerable range.
Because optimization methods require a relatively simple description of
a system,it is advisable to test the policies selected by optimization models
with simulation models which incorporate a more detailed description of the
system. Holling and Dantzig (1978) tested the policy generated by the Dantzig-
Winkler algorithm with the simulation model developed by Jones (1977). The
results of the spatially disaggregated simulation model indicated that profits,
total wood supply and employment over a one hundred year planning horizon are
greater with the Dantzig-Winkler policy than with the policy which has
historically been used in New Brunswick. Although the Dantzig-Winkler policy appears
to perform better than the current policy, it is unlikely to be a true optimum
since the optimization formulation could not incorporate a realistic
description of the impact of the age composition of trees and spatial heterogeneity of
the pest density.
In order to overcome some of the simplifications associated with the
Dantzig-Winkler policy, Stedinger (1977) developed an alternate formulation.
Initially Stedinger considered a single site of trees of mixed ages under
steady-state conditions. He assumed the age distribution of the trees was
stable, jl._e., that 1/60 of the forest consisted of trees which were i years
old, i = 1, ..., 60. To minimize the infinite-horizon, discounted costs of
pesticide spraying, Stedinger solved the following recursive dynamic
programming equation:
L
H (s) = Min [R(s, v) + a I P(o).|s, v) H (a))] (3)
v(s) ~ ~ i=l -1' ~ ~
where s. is the egg mass density and s« is an index of deformation. Stedinger
calculated the transition probabilities P(o) |s, v) from his own detailed
simulation model of spruce budworm population dynamics in Maine. He used Howard's
(1960) policy iteration procedure to compute the solution to (3).
In order to insure that the stable age distribution of trees was
maintained, Stedinger could not allow tree mortality to occur; hence he constrained
v to equal one when egg densities or defoliation are sufficiently high. This
is a very restrictive assumption since it may be more economical to let trees
die in some areas rather than to spray them year after year. Stedinger also
compared the output of single site and multi-site models for a variety of
control strategies. He concluded that the single site management model used both

158
R.B. Rovinsky and C.A. Shoemaker
by himself and by Winkler (1975) is not adequate because it ignores the
economic value of spraying moderate or high density populations in order to
prevent infestation of adjacent sites.
In many cases, especially in control of insect pests of field crops, it
is very important to incorporate dynamic changes in the age structure of the
pest population into pest control decisions. Age distribution is significant
because the pest's susceptibility to pesticide, its ability to inflict crop
damage and its reproductive potential all depend upon the age of the pest.
Inclusion of age class sizes in the state vector of an optimization problem
usually increases the dimension of the problem so much that it is not
computationally feasible to calculate the optimal solution. This is similar to
the problem of describing the age structure of host trees encountered in the
studies of spruce budworm.
Shoemaker (1979) suggests a method of circumventing the dimensionality
problems associated with the incorporation of pest age structure into a dynamic
programming formation. Her approach is to use the state vector to describe the
times of previous pesticide applications. The recursive dynamic programming
equations are in the form of Equation (2) with the state vector s = (s , s )
giving the times of the two most recent pesticide applications. The function
R is the negative of the cost during period k of crop damage and pesticide
applications. The amount of crop damage occurring in a period depends upon
the age structure of the population which is computed at each stage from the
equation
X. = S(k-i, s)A(k-i)
where X. is the number of pest3 in the i age class in period k, A(k-i) is the
number of individuals recruited at time i and S (k-i,s) is the fraction of the
cohort which survive from period k-i to period k given that the most recent
periods of pesticide application are at times s = (s , s ). Expressing
survival S as a function of s is an approximation since it is possible that more
than two pesticide applications have been made. However, because pesticide
mortality rates are quite high, the fraction of the population which would
survive two pesticide applications is quite small. Therefore, the actual
survival can be closely approximated by considering only the two most recent
pesticide applications. As a result the state of the system can be
characterized by only two variables.
Because of this two dimensional characterization of the pest population,
the optimal timing of pesticide applications can be computed more efficiently
for population models with a large number of age classes than is possible with
the methods used previously. The two dimensional characterization is based

APPLICATIONS IN AGRICULTURE
159
on a number of assumptions including restrictions that survival and damage are
deterministic functions and that seasonal damage can be represented as a sum of
the damage occurring in each stage. It is also possible to incorporate
nonlinear damage or stochastic effects by inclusion of an additional state vector
(Shoemaker, in press). Solution of the associated three or four dimensional
dynamic programming problem requires more computation time than in the linear,
deterministic case, but even for a large number of age classes these
computations are quite feasible.
Another paper by Shoemaker (1977) develops a procedure for solving an
optimization model which incorporates biological and cultural pest control
procedures as well as the use of pesticides and a dynamic description of pest age
structure. The inclusion of biological control provided by an insect parasite
considerably complicates the calculation of optimal policies because pesticide
and cultural control procedures may kill parasites as well as pest insects.
The approach used by Shoemaker is to develop two nested models. The first
model is a multi-year, stochastic dynamic programming model with three state
variables. The recursive equations of this model have the form of Equation (1)^
where the decision vector v describes the times of harvest (v..) and the dosage
of pesticide application (v9). The components of the state vector are the
number of pests (s..) and parasites (s_) at the beginning of the season and the
weather pattern which is described by a random variable s~. The variable s
is important because synchrony between the crop, the pest and parasite
populations can be significantly altered by changes in temperature.
The second model is a more detailed population model which is used to
k k
estimate the effect of management decision v and state vector s on yield
k+1 k+1
losses and on the size of next year's populations (s , s9 ). The population
model incorporates the description of the effect of temperature on dynamic
changes in pest age structure, on crop susceptibility and on synchrony between
the occurrence of parasites and susceptible pest age classes. A detailed
population model describing these interactions could have been simulated to
k k k+1
calculate the decision model's transition functions G-(s , v ) = s. and
k k k+1 "~
G?(§ , y ) s s« and the return function R(s, y). Unfortunately there are so
many combinations of values of (s, y) that simulation of the population model
for each value of (s, v) is not feasible.
To avoid the computational difficulties associated with trying to compute
a simulation model for each value of s and y, Shoemaker developed a differential
equation model which mimieked the most essential features of existing simulation
models. The advantage of the differential equation model is that it is
possible to solve parts of it analytically.

160
R.B. Rovinsky and C.A. Shoemaker
The basis of the population model is an equation describing the effect of
predation and other mortality factors on the number of unparasitized insects
(N (r,b)) born at a particular time (b) and surviving to at least level r of
n °
maturation
dN
-r-^ (r,b) = - D(r-b)N (r,b) - a(r-b)N (r,b)P(h(r))h(r) (A)
ar n n
where D is the death rate, P(t) is the number of parasites attacking at time t
and a is the rate of attack. The number of active parasites at any point in
time is P(t) = s!rP (t) where s« is the total number of parasites and P (t)
z o z o
describes their distribution through time. The parasite only attacks certain
age classes so a has the form
a (m) = o^ if A± <_m £A2 (5)
= 0 otherwise
where m is the maturity of the individual. The pest population matures at a
rate that is dependent primarily on temperature. The activity of the parasite
is more closely related to the passage of calendar time. The index t is used
to denote the maturation time scale of the parasite and r refers to the
maturation scale of the pest. The function h(r)=t relates the two time scales.
Computation of h(r) depends upon temperature, and it is defined in such a way
that the maturity m of N (r,b) isr-b. The term h (r) is required in Equation (U)
to convert the parasitism rate a (which is in units of t) to a rate in terms
of units of r.
Following a similar argument, the number of parasitized insects is
dN
, (r,b) = -D(r-b)N (r,b) + a(r-b)N(r,b)P(h(r))h(r) . (6)
ar p
Since all eggs are unparasitized, N (b,b) = 0 for all b. The number of new
eggs being laid, N (b,b), is specified in an input function 9 which describes
n k k
the distribution of oviposition over time, jL._e. , N (b,b) = s 0(b)/2.
The solution to Equation (A) is
n n
b
(r
- 1 [D(T-b) + a
N (r,b) =N (b,b)exp[- I [D(T-b) + a (T-b)P(h(x))h(T) ]dT ] (7)
k W _ ,_w„_u^™,_a J P
= s^ Zf^ exp (-K(s-b))exp(-a1 J P(h(s))h(s)ds) (8)
where m
K(m) =
D(T)dT , (9)
_r(b,r) = Min(b+A1,r) and r(b,r) = Min(b+A2>r),

APPLICATIONS IN AGRICULTURE
161
The interval (£(b,r), r(b,r)) is the period preceding r when the individual
isceptible to paras
(8) can be replaced by .
is susceptible to parasitism, jL.£, when a(r-b) = a . The integral in Equation
L(b,r,s2,s3) = P(t)dt (10)
t_
where t = h(r(b,r)) and t_ = h(£(b,r)). The function h(r) is determined by the
random state variable s~ which depends upon weather.
From the equations above, it follows that in the absence of pesticide
applications
Nn(r,b) = s* -2M R(r-b)exp {-^(b.r ,s2,s3)} r^b (11)
where R(a)
K (T)dT.
o
Harvest and pesticide mortality is modelled by
N (vj,b) = <Kv9,vrb)N(v~b) (12)
n 1 z 1 n 1
where v is the time of harvest, \?2 is t5ie insecticide dosage, and 4> (v,m) is
the survival from pesticide and harvesting. Then the general expression is
N (r,b) =N (b,b)R(r-b)3l(v,b,r)exp{-aiL,(b,r,s-,sQ)} (13)
n n 1 Z J
where Lf is adjusted to include the effects of pesticides on parasites and
tf(v,b,r) = ^(v,b) if r > v1
= 1 otherwise (14)
The dynamic programming model requires evaluation of the functions G.. and
G? which describe year-to-year changes in the values of the state vector. The
k+1
number of pests (s ) at the beginning of year k+1 is the fraction (y ) which
1 w
survive the overwintering period multiplied by the number which have survived
to the end of (r£) of the previous growing season, which is I N (r£,b)db.
Combining the relationships described above, we can show that
k+1 k „, A
sn = s R(Amax)y
1 X G
M(b,v)exp{-a1L'(b,r,S2>s^)} dh
G1(s1,s2,s3,v1,v2) (15)
where
M(b,v) --2M rf(v fv -b) (16)

162
R.B. Rovinsky and C.A. Shoemaker
and A is the age at which pests enter their overwintering stages. Similarly
max
the number of parasites in the following year is
_k+l _ k+1 „ ,A ,
S0 = s. R (A )u
2 X 2 P p
k k
Mf(b,v)[l-exp{-a1Lf(b,r,s2>s3)l db
= G2(sk,vk) (17)
where
Mf(b,v) = M(b,v) if v - b < A
= y-^- otherwise (18)
The scalar A is the age at which parasites kill their hosts and become pupae.
P
The pupae are not susceptible to pesticide but a fraction Y will be killed by
harvesting. The scalar y is the fraction of parasites which survive the
winter.
Since the function L and the integrals must be solved numerically,
Equations (15) and (17) do not represent an analytical solution. However,
using Equations (4) and (5), a highly efficient numerical procedure was
developed which loops through the values of s , s9» s~, v and v9 in such a way
that solutions for one set of values can be utilized in computing G and R
for other sets of values. A description of this procedure and its
application to alfalfa weevil management is given in a paper by Shoemaker (in press).
The Optimal Distribution of Crop Production Under
Differing Levels of Farmer Cooperation: An Example Using Cotton
An important policy problem in the economics of pest management is how to
compute the set of economic equilibria that results from differing policy
assumptions. This is particularly important when trying to predict
agricultural production and distribution resulting from a change in pesticide use or
policy. A variety of linear, quadratic, and concave programming methods have
been developed to determine, for a variety of agricultural policies, the
optimal location of crops, livestock, and producers (Duloy and Norton, 1975;
Hall et. al., 1975; Heady and Srivastava, 1975; Takayama and Judge, 1971; Von
Oppen and Scott, 1976). However, previous studies in this field assumed only
one type of competition among farmers; either farmers operate only in pure
competition as price-takers, or they are assumed in total cooperation. Further,
present algorithms require for their solution that the objective function or
decision criterion be a concave or convex function. Finally, most policy
models in developed nations have been concerned with farm inputs, crop
production, or land and water use, and have concentrated on the feed grain and food

APPLICATIONS IN AGRICULTURE
163
sector. They exclude as exogenous cotton production, even though over 50% of
insecticide use in the U.S. is devoted to this crop. By using a fixed
national demand for cotton lint, the models limit cotton land shifts both
nationwide and within individual crop regions as well.
Because of these limitations, these models cannot consider such policy
issues as changes in insect pest management strategies, irrigation
practices, cotton support and set-aside programs, or competition from synthetic
fibers. Since it is impossible to eliminate a single fixed demand without
considering some type of price structure to control and determine supply, it
is necessary to examine models in which both production and price are at
least partially variable. Limited work in this area has been done by Casey
and Lacewell (1973) and Evans and Bell (1978).
In what follows we present methods and results that relax the constraints
discussed above, and greatly extend the methods and applications. We develop
and use a new algorithm for solving a particular class of nonconvex quadratic
programming problems, which has proven to be quite efficient and inexpensive,
and show applications to a recent insect pest management study (Pimentel etal.
1979).
To begin, consider a model where a large number of producers are
aggregated into N subsets that we will call regions. Producers in region j can
produce either a commodity C (in the study cited above, this is cotton) or an
alternative commodity A. (in the study, soybeans or grain sorghum) whose price
is assumed independent of the actions of all other producers. The price of C
in region j depends upon the total national production of C, but the regional
price of the alternative commodity A. is assumed independent of the actions
of all other producers. This latter assumption is reasonable in this analysis,
because the national acreage of cotton is much less than the total acreage of
soybeans and grain sorghum, and a majority of acres in these crops are planted
outside the principal cotton producing states.
The equilibrium allocation of resources depends upon the amount of
cooperation among individual producers and among regions. Let T be the total
production of commodity C when each individual producer acts independently as
a price-taker to maximize his profit. Let T be the level of production of
commodity C resulting when producers within a region cooperate to maximize
Q
profit. Finally, let T be the level of production of commodity C when all
producers in all regions cooperate to maximize total global (national) profits.
L R G
The vectors X , X and X give the amounts of commodity C produced in each
region for each of these situations.

164
R.B. Rovinsky and C.A. Shoemaker
Mathematical Results
We have elsewhere (Rovinsky, 1977; Rovinsky et al., in press) presented
L R G
results which yield methods for calculating X , X , .and X , These results will
be briefly stated and reviewed below.
First, we define the following for each region j:
P. - price of commodity C
x. - the number of units of C produced
L. - total capacity
s. - profit per unit of capacity for producing alternative commodity A.
y. - amount of C produced per unit of capacity
b. - the maximum production of commodity C (b. ■ L.y.)
J 3 3 3
c - production cost of C per unit of capacity.
Let T denote the total production £ x. of commodity C and assume that T
j 3
determines the price in each region by the simple linear relationship
P. - d. T + f.
3 3 3
where d <0 and f. are constants. Then the profit per unit capacity in region
j obtained by producing commodity C is (d. T + f.)y. -c.. The profits per
unit of capacity are equal for the two commodities when T = q. where
q. = -m./d. and m. * f. - (c. + s.)/y. (2)
3 3 3 3 3 3 3 3
Producing commodity C in region j is more profitable than producing commodity
A. if and only if T < q..
3 J L
Result 1: To find X , order the regions so that qn > ... >qXT. Let k be
k 1- - N p
the smallest k so that ? , b.> q. . If there is no such k, set k » N+l. Then:
3=1 3 k P
„L
f
X, - < max (0, q. - ,Z. b.)j«k (3)
j
b, 1 < j < k
J k-1 ~ p
P
*kp j-1 j/J p
\ 0 k^< j < N
P
This algorithm is due to Casey and Lacewell (1973). We have shown that,
if each producer is acting independently to maximize his profit, then the
distribution of production X given above is an economic equilibrium; i.e., no
producer can independently act to increase his profits. Further, it is the

APPLICATIONS IN AGRICULTURE
165
unique such equilibrium.
Result 2: Under regional equilibrium, the total profit in region j is:
N
(d. ±£l xi)Xj + m.Xj + s.L. (4)
To find XR, define
f(T) = l{b±: 0<T<q^ bi>+ E{q± - T: q. - b. < T < qJ . (5)
R R
Then it is easy to see that f has a unique fixed point T , and X is given
by
if 0 <TR <q. - b.
3 J
x* = { qj-TR if q. -bj <TRlq. (6)
otherwise
We have also shown that X is the unique regional equilibrium, that the
N R R R L
resulting total production .£ x. = T , and that T <_ T .
Result 3: If we assume cooperation among regions as well as among
producers within a region, then the total profits can be further increased. From
(4), we note that the total income from commodities C and A , A9,...,AT is
N N N N
.1. .£- d.x.x. + £ m. x, + E s.L., ,
i=l j=l i i J i»l i i i=l i i (7)
and thus the problem of finding a global equilibrium can be stated as the
following nonconcave quadratic programming problem:
N N N
max 8<x) = fa fa d.x^. + fa m±x± (8)
subject to 0 <_ x <_ b , j-1, . . . ,N.
3 ^
We have shown that if T=£x., r. = d.T + m , and the regions ordered for each T
so that r > ... _> rN, then we can easily define an optimal x*(T), by a method
similar to the Casey-Lacewell ordering algorithm above. Thus we redefine our
problem above to the one-dimensional
max 0(T) = 6 (x*(T)) (9)

166 R-B- Rovinsky and C.A. Shoemaker
N
0 < T < £ b.
- ~ J = l J
N
We have shown that 0 is continuous on S = [0, £ b.], piecewise strictly
concave, and twice differentiable on all but a finite, known subset of S. Thus we
Q
are led to a fairly straightforward search for T in S. The search is
significantly faster than the existing nonlinear programming algorithms. Details
of the algorithm and a listing of the FORTRAN IV program written to implement
it are given in Rovinsky (1977). This algorithm, denoted THETA, also finds
the local optimal and regional equilibrium solutions described above.
We can significantly reduce the size of the interval of search by finding
Q
upper and lower bounds for T . We have shown that if T* is any local maximum
R G R G
of 0, then T* £ T . In particular, T <_ T for any T which maximizes 9(T)
Q
for T £ S. A lower bound T . has also been found for T , and THETA calculates
G mln ' *
for T by finding and comparing all local maxima T_ between max(0, T . ) and
R G * G mln
T . The optimal distribution X is then set equal to x (T ).
As discussed above, the THETA algorithm was developed to allow changes in
prices, demands, yields, and costs of cotton production to be incorporated into
large scale policy models. Thus, instead of a set of fixed demands, price and
demand could vary, and regional production could shift accordingly. The THETA
algorithm solves for the equilibrium solutions very inexpensively relative to
the size of the problems. For example, these methods were applied as part of a
recent study of insert pest management methods. Cost, yield, and price data for
cotton and its alternate crop? were gathered for a partition of the Southern
United States into 300 regions. Cotton production and total farm income were
computed for the three levels of producer cooperation discussed above, and for
each of fifteen price elasticities used to simulate the short, medium, and long
run equilibrium estimates for cotton demand. Less than one minute of IBM 370/168
CPU time was required to calculate the complete set of solutions. This
algorithm has been extended to a wider class of nonconcave quadratic
programming problems. The critical assumption that revenue per unit of capacity is
constant for the alternate commodities may be viewed as assuming that the value
of unused capacity is fixed. This constraint can be partially relaxed by
placing restrictions on shifts of regional production and/or return to C and the
alternate commodities. This allows consideration of those situations in which
an existing production distribution may be modified under constrained
guidelines.
Applications to National Pesticide Policy Analysis
Starting in the late 1960fs, researchers began studying the impact of
various pesticide policies upon farm income and production. Early work
focussed on possible Environmental Protection Agency (EPA) restrictions on or

APPLICATIONS IN AGRICULTURE
167
cancellations of widely used insecticides such as DDT, dieldrin and aldrin.
Currently the Department of Agriculture (USDA) responds to an EPA notice that
a certain pesticide's registration (or re-registration) is being reviewed, by
supplying data on the benefits and costs of the pesticide's use by farmers.
At first USDA used a partial budgeting approach. This involved first polling
state entomologists and economists to obtain the relative inputs of the
affected pesticide and its alternatives in their areas. Then the cost and
yield changes obtained were aggregated to derive regional and national effects.
This approach drew criticism for its failure to reflect expected adjustments by
producers and the market (shifts in crop acreage and prices) in response to the
proposed pesticide cancellation.
In the early 1970*s, several researchers outside USDA began using
existing linear programming (LP) models to examine the effects of pesticide
policies. Such models assumed the farm acreage in the country could be divided
into several hundred homogenous regions. The cost of production and yield for
a given crop were assumed to be constant throughout each region. The decision
variables described the number of acres in each region which were devoted to
each crop. Additional variables described the amounts of each crop transported
to other regions to meet market demand. In order to investigate the spatial
effects of a pesticide ban, yield and cost coefficients in .the linear
programming model were changed to evaluate the cost and effectiveness of pest control
strategies which would be substituted for application of the banned pesticides.
Then the base LP solution (i.e. that solution obtained by solving the model
under present conditions) could be compared against a proposed pesticide ban
solution to examine regional shifts in production and acreage. Various
accounting schemes can then be used to examine environmental and economic effects.
The primary source of LP models and data was a series of studies
beginning in 1954 at Iowa State University under the direction of Earl 0. Heady
(Heady and Srivastava, 1975). Heady1s work established these models as
legitimate policy tools, but they proved to be large, not well documented, and
expensive, and several smaller variations were developed. All were cost-
minimization LP models, with variable costs and fixed demands for food, feed,
and fiber. Output included transportation and production costs, acreages, and
production by crop and region. Their solutions were interpreted as
projections of cropping patterns to the year 1980, 1985, or 2000. They generally
showed large shifts of production, emphasizing the decreasing and westward
moving trends in agricultural land use.
Several problems arose in using these LP models, and researchers and
policy analysts were generally unwilling to accept the massive interregional
shifts of production predicted. Modifications of the algorithms were made to

168
R.B. Rovinsky and C.A. Shoemaker
constrain acreage shifts using "policy" upper and lower bounds. Penalties
were attached to gross shifts from current levels to simulate farmers'
reluctance to change, and regions were further disaggregated by soil quality.
Current work is directed towards meeting another problem with LP, the absence
of demand functions. The use of quadratic programming and the implanting of
stepped demand functions have aided the process, as we have discussed in the
previous section.
In 1977 the Natural Resources Economics Division (NRED) within USDAfs
Economic Research Service began to collect and analyze the current state of
pesticide models and data. Their goal, in using models, was to increase the
reliability of their short term research results on the economic consequences
of continuing or discontinuing the use of particular pesticides for particular
purposes. They hoped also to decrease the number of hours required to estimate
the benefits of each use of a pesticide. Finally, they expected to test the
suitability of each model for future pesticide policy analyses.
This effort attracted the interest of many modelers outside USDA, and
generated many proposals to adapt a variety of LP, simulation, and econometric
models to analyze pesticide policy. The economists in NRED decided to pursue
a 3-stage plan: first to use current, inexpensive working models or approaches;
then to adapt current, large models; and finally to investigate possible
linkages and improved models. Each model was run using a common data base.
This consisted of a set of cost and yield changes, by state, resulting from
a proposed EPA cancellation of a leading cotton insecticide.
There were 3 initial models: (1) a static partial budgeting accounting
system; (2) PESTDOWN, a simplified linear programming model with 8 crops, 135
producing regions and A soil classes/region; and (3) POLYSIM, an econometric
model built around the USDA 5 year baseline economic forecasts. These
approaches met the criteria of data and model accessibility, low cost, and
ease of use. The second stage used four models: the current Iowa State
University linear programming model; a stochastic version of POLYSIM; a
commercial Macro-Econometric model; and a cotton model constructed at USDA.
In what follows we briefly describe the structure of each of the models. For
details and a full description of how each model was used see Rovinsky et al.
(1979).
Both the ISU and PESTDOWN models are total variable cost minimization
Linear Programming (LP) models of the following general form:
(P) minimize F = ex
so that A-X >^ D
A2X £ R

APPLICATIONS IN AGRICULTURE
169
where the vector c contains the variable cost of production for each cropping
activity, the costs of transporting farm commodities between demand regions,
and other model specific costs; the vectors D and R contain the exogeneously
determined demands and resources, respectively, and the matrices A. and A_
contain the technical coefficients of production and transportation. The
models are both quite large and very expensive to solve initially; thus,
subsequent policy runs are made using the base solution as a starting point.
This reduces the cost considerably.
Both models contain the same set of seven crops: corn, sorghum, wheat,
oats, barley, soybeans, and cotton. The ISU model also includes activities
for hay and silage, while PESTDOWN includes rye and peas in its allowable crop
mix. Using the FEDS budget generator at Oklahoma State to supply cost and
yield data, production activities are defined on 135 producing regions in
PESTDOWN and 105 producing areas in the ISU model. Demands for agricultural
commodities are described for a set of "consuming" regions, which in PESTDOWN
are aggregates of States and in the ISU model are general market areas around
principal trade cities.
Exogenously determined changes in production costs and crop yields,
resulting from a given pesticide policy, are used to change the base cost and
crop yield values utilized in the models. Each change in cost must be expressed
in base-year dollars and enters the model as an amount to be added to (or
subtracted from) the per acre variable costs of producing a given crop in a
given region. Each change in yield must be expressed as a ratio of new
regional costs and yields* the model will determine the optimal (cost-minimizing)
solution for the new scenario. The solution's output describes total and
regional acreage and production by commodity. The impacts of a pesticide ban
may then be estimated through a comparison of the base and cancellation runs.
The econometric cotton model is one of several commodity-specific models
developed within USDA. It was designed for use in evaluating the effects of
policy changes on the U.S. cotton market. The model consists of three basic
sections; a price block; a supply block; and a demand block. Each block
consists of a set of equations and mathematical identities. There is a total
of 30 equations and five identities which exactly determine the 35 endogenous
variables, and the model contains approximately 70 exogenous variables. The
supply block is separated into four cotton production regions; price variables
are common to the supply and demand segments of the model.
The model simulates annual behavior in the cotton market. The
coefficients of the model were estimated using ordinary least squares regression
over the period 1951 to 1976. The model consists of a system of non-linear
equations and is solved using the Gauss-Seidel gradient method. Levels of

170
R.B. Rovinsky and C.A. Shoemaker
demand and supply and endogenous cotton price variables are simultaneously
determined for each given year. Within blocks of the model, however, equations
are solved in a recursive fashion, ending with the determination of a cotton
market equilibrium. Exogenously determined values for the regional changes
in production costs and yields are used to change key variables in the model
for a pesticide policy impact run. These changes enter the model through the
regional yield and planted acreage equations.
The cotton model provides five year forecasts of a number of important
cotton sector variables. If run twice, once as a base and once after
incorporating policy induced changes in efficiency and costs of cotton production,
that policy's impact on cotton yield per acre, regional and total planted
acres, regional and total production, farm and wholesale price, foreign
exchange earnings from cotton exports, total consumption, and consumers' and
producers' surpluses may be estimated.
POLYSIM was constructed differently from most economic or econometric
simulation models to ensure compatibility with other policy analyses of USDA.
The model makes full use of independent forecast data as a reference baseline.
Included are the five-year baseline projections of commodity supplies, prices,
and utilization made by USDA commodity specialists using formal forecasting
models tempered with their own experienced judgments. The projections contain
explicit assumptions concerning the rates of change in population, per capita
income, consumer preferences, export demand, technology (including crop yields
and livestock gains), and other supply and demand shifters. These projections
also assume a specific set of agricultural programs.
POLYSIM is a recursive economic simulation model. Ceteris paribus, supply
is a function of the previous year's price; the current year price is determined
by the level of supply relative to expected demand; and actual quantity
demanded is a function of price. The driving mechanisms in the POLYSIM model
are the initial and subsequent changes in commodity prices resulting from
policy changes. The magnitude of impact is determined by each commodity's
own price elasticities, as well as cross supply and demand elasticities.
Commodities included in the model are feed grains, wheat, soybeans, cotton,
cattle and calves, hogs, sheep and lambs, chicken, turkeys, eggs, and milk.
As indicated earlier, the model is designed to simulate around a set of
baseline estimates; these must be available for all years analyzed. To date,
most applications have been for a time horizon of three to five years.
Government economists also utilize models developed by the private sector
to conduct their research. These include the commonly used products that are
commercially marketed by three firms—Chase Econometrics Associates, Inc.
(Chase), Data Resources, Inc. (DRI), and Wharton Econometric Forecasting

APPLICATIONS IN AGRICULTURE
171
Associates, Inc. (Wharton). Typical products include all of the following:
access to one or more models, data bases and econometric software; receipt of
economic and financial reports and forecasts; and some consulting. Although
the commercial models are different from each other in many ways, there are
enough similarities so that the following discussion of the Chase system will
give a general idea of what is available elsewhere.
The two components of the Chase Econometric System most often used for
pesticide analyses are the Chase Agricultural Model and the Chase Macroeconomic
Model. Changes in base yields and costs of production due to a pesticide
regulation are used for input data to the Agricultural Model. These produce
output on changes in farm level prices, crop acreage, production, etc.
Changes in farm level prices obtained from the Chase Agricultural Model, or
other models (such as the CED Cotton Model or POLYSIM which are discussed
above) are then used to shock the Chase Macroeconomic Model; these macro-
economic simulations indicate the influence of agricultural events on the
general economy. Macroeconomic performance indicators include WPI (the
Wholesale Price Index), CPI (the Consumer Price Index), GNP (Gross National
Product), etc.
The USDA researchers found, as expected, that the econometric simulation
models tended to be less expensive, and more acceptable to other economists,
who tended to be suspicious of the LP model's lack of prices and copious
detail. They also discovered that, with the possible exception of their
Cotton model, there was no adequate regionalization in the econometric models,
to reflect yield variations and shifts over time. Perhaps most importantly,
their work has stimulated a complete review of their cost and yield data,
especially that required for the LP models, for which inaccuracies could cause
the model to put all production in one place. Through extensive use of the
commodity models, they have encouraged the development of the more integrated,
cross-commodity models. Finally, their work was used in a recent draft study
on the feasibility of eradicating the Cotton Boll Weevil.
Current work at USDA is aimed at two objectives. There is interest in
attacking a wider variety of pest policy problems other than a single
pesticide ban. The range of models available, and the fact that they are constantly
being improved by their developers, may make it possible to study a range of
integrated pest management techniques, and to combine a controlled set of
pesticide applications, cropping techniques, early scouting for pest problems,
and careful plant breeding to improve pest control and decrease pesticide use.
There is also an attempt underway to link the optimization and simulation
models, hopefully emerging with a better forecasting and analysis tool. "Rules
of thumb," which apply the percentage effects of the normative (LP) models to

172
R.B. Rovinsky and C.A. Shoemaker
the base output of the positive (econometric) models, have been proposed, as
well as the use of updated sets and stepped demand models.
REFERENCES
1. Agrawal, R.C. and Heady, E.O. , Operations Research Methods for
Agricultural Decisions, Iowa State University Press, 1972.
2. Bahrami, K., and Kim, M., "Optimal Control of Multiplicative Control
Systems Arising From Cancer Therapy," IEEE Trans. Automatic Control, pp. 537-
542, 1975.
3. Bellman, R.E. and Dreyfus, S.E., Applied Dynamic Programming, NJ,
Princeton University Press, 1962.
4. Bellman, R.E. Dynamic Programming, NJ, Princeton University Press,
1957.
5. Bellman, R.E. and Kalaba, R., "Some Mathematical Aspects of Optimal
Predation in Ecology and Boviculture," Proceedings of the National Academy of
Sciences, vol. 46, pp. 718-720, 1960.
6. Casey, J.E., Jr., and Lacewell, R.D., "Estimated Impact of
Withdrawing Specified Pesticides from Cotton Production," Southern Journal qf
Agricultural Economics, vol. 5, No. 1, 1973.
7. Casey, J.E., Jr., and Lacewell, R.D., "A General Model for Estimating
the Economic and Production Effects of Specified Pesticide Withdrawals: A
Cotton Example," Texas A&M University, Environmental Quality Program, Note 15,
1974.
8. Comins, H.N., "The Management of Pesticide Resistance," Journal of
Theoretical Biology, vol. 65, pp. 399-420, 1977.
9. Conway, G.R., "Mathematical Models in Applied Ecology." Nature.
269(22), 1977.
10. Dantzig, G.G.9 "Determining Optimal Policies for Ecosystems,"
Technical Report, No. 74-11, Stanford University, California, 1974.
11. Day, R.H. and Sparling, E., "Optimization Models in Agricultural
and Resource Economics," A Survey of Agricultural Economics Literature, vol. 2,
edited by G.C. Judge et al., University of Minnesota, 1977.
12. Duloy, J.H., and Norton, R.D., "Prices and Incomes in Linear
Programming Models," Revised Version of Discussion Paper No. 3, Development Research
Center, International Bank for Reconstruction and Development (World Bank),
New York, 1975.
13. Evans, Sam and Bell, Thomas M., "How Cotton Acreage, Yield, and
Production Respond to Price Changes," Agricultural Economics Research, vol.
30, No. 2, April 1978.
14. Fick, G.W., ALSIM I (Level 1) User's Manual, Cornell University,
Department of Agronomy, Mimeo 75-20, 1975.

APPLICATIONS IN AGRICULTURE
173
15. Greenberger, M., Crenson, M.A., and Cressey, B.L., Models in the
Policy Process. Russell Sage Foundation, New York, 1976.
16. Hall, H.H., Heady, E.O., Stoecher, A., and Sposito, V.A., "Spatial
Equilibrium in U.S. Agriculture: A Quadratic programming Analysis," SIAM
Review,17, 2, 323-338, 1975.
17. Heady, E.O., and Srivastava, U.K., Spatial Sector Programming
Models in Agriculture. Iowa State University Press, Ames, 1975,
18. Holling, C.S. and Dantzig, G.B., "Determining Optimal Policies for
Ecosystems: An Overview of a Case Study," Technical Report, Institute of
Resource Ecology, University of British Columbia, Vancouver, B.C., 1978.
19. Holling, C.S., Jones, D.D., and Clark, W.C., "Ecological Policy
Design: A Case Study of Forest and Pest Management," Pest Management, G.A.
Norton and C.S. Holling (eds.) London, Pergamon Press, 1978.
20. Howard, R., Dynamic Programming and Markov Processes, NY, Wiley,
1960.
21. Hueth, D., and Regev, U., "Optimal Agricultural Pest Management with
Increasing Pest Resistance," American Journal of Agricultural Economics, vol.
56, pp. 524-552, 1974.
22. Huffuker, C.B., New Technology of Pest Control, NY, Wiley (in
press).
23. Jaquette, D.L., "Mathematical Models for Controlling Growing
Biological Populations: A Survey," Operations Research, vol. 20, pp. 1142-1151,
1972.
24. Jones, D.D., "The Application of Catastrophe Theory to Ecological
Systems," Simulation in Systems Ecology, G.S. Innis (ed), Simulation Council
Proceedings, 1977.
25. LaDue, E.L., Shoemaker, C.A., Russell, N.P., Rovinsky, R.B., and
Pimental, D., "The Potential Impact of Cotton Insect Control Technology,"
Cornell Agricultural Economics Staff Paper No. 79-31, Ithaca, 1979.
26. Petit, M., "The Role of Models in the Decision Process in
Agriculture," Decision Making and Agriculture, edited by T. Dams and K.E. Hunt,
University of Nebraska Press, 1977.
27. Pimentel, D., Shoemaker, C.A., Ladue, E.L., Rovinsky, R.B., and
Russell, N.P., "Alternatives for Reducing Insecticides on Cotton and Corn:
Economic and Environmental Impact," Report to the Environmental Protection
Agency, 1979.
28. Rausser, Gordon C, Just, Richard E., and Zilberman, Davis,
"Prospects and Limitations of Operations Research Applications in Agriculture
and Agricultural Policy," California Agricultural Experiment Station Giannini
Foundation of Agricultural Economics, Berkeley, March 1980.
29. Regev, U., Gutierrez, A.P., and Feder, G., "Pests as a Common
Property Resource: A Case Study of Alfalfa Weevil Control," American Journal
of Agricultural Economics, vol. 58, pp. 188-196, 1976.

174
R.B. Rovinsky and C.A. Shoemaker
30. Rovinsky, R.B., Shoemaker, C.A., and Todd, M.J., "Determining Optimal
Use of Resources Among Regional Producers Under Differing Levels of Cooperation,"
Operations Research, Vol. 28, No. 4, July-August 1980.
31. Rovinsky, R.B., Reichelderfer, K.H., Weisz, R.N., and Quinby, W.A.,
"The Use of Policy Models to Evaluate the Effects of a Pesticide Ban," 1979
(to appear).
32. Shoemaker, C.A., "Optimal Integrated Control of Pest Populations
With Age Structure," Operations Research (in press).
33. Shoemaker, C.A., "Optimal Timing of Multiple Applications of Pesticides
With Residual Toxicity," Biometrics, December 1979.
34. Shoemaker, C.A., "Optimization of Agricultural Pest Management III:
Results and Extensions of a Model," Mathematical Biosciences, vol. 18, pp. 1-22,
35. Shoemaker, C.A., "Deterministic and Stochastic Analyses of the Optimal
Timing of Multiple Applications of Pesticides with Residual Toxicity" Proceedings
of Pest Management Modelling Conference« John Wiley and Sons, New York (in
press).
36. Shoemaker, C.A., "Pest Management Models of Crop Ecosystems,"
Ecosystem Modelling in Theory and Practice, edited by Charles A.S. Hall and
John W. Day, Jr., John Wiley & Sons, N.Y., 1977.
37. Stedinger, J.R., "Spruce Budworm Management Models," Ph.D. Thesis,
Department of Applied Physics and Engineering, Harvard University, Cambridge,
1977.
38. Takayama, T., and Judge, G.G.9 Spatial and Temporal Price and
Allocation Models, North-Holland Publishing Co., Amsterdam, 1971.
39. Taylor, C.R., and Headly, J.C., "Insecticide Resistance and the
Evaluation of Control Strategies for an Insect Population," Canadian Entomologist,
vol. 107, pp. 237-242, 1975.
40. U.S. Department of Agriculture, "Agricultural Statistics," U.S.
Government Printing Office, Washington, D.C. 1975.
41. Von Oppen, M., Scott, J., "A Spatial Equilibrium Model for Plant
Location and Interregional Trade," American Journal of Agricultural Economics,
58, 3, 437-445, 1976.
42. Watt, K.E.F., "The Use of Mathematics and Computers to Determine
Optimal Strategy and Tactics for a Given Insect Control Problem," Canadian
Entomologist, 96, 1964.
43. Waugh, F.V., "Demand and Price Analysis - Some Examples from
Agriculture," Technical Bulletin No. 1316, Economic Research Service, U.S.
Department of Agriculture, Washington, D.C. 1964.
44. Wickwire, K., "Mathematical Models for the Control of Pests and
Infectious Diseases: A Survey," Theoretical Population Biology, vol. 11,
pp. 182-238, 1977.
45. Winkler, C, "An Optimization Technique for the Budworm Forest-Pest
Model," International Institute for Applied Systems Analysis, RM-75-11, Laxen-
burg, Austria, 1975.

Proceedings of Symposia in Applied Mathematics
Volume 25, 1981
MATHEMATICAL MODELING APPLIED TO THE
RELOCATION OF FIRE COMPANIES
Warren E. Walker
The Rand Corporation
ABSTRACT. In recent years fire departments in urban areas have
experienced a sharp increase in demands for their services while their
budgets have generally grown at a rate less than that of inflation.
Many of these departments have turned to systems analysis for help,
realizing that, if they do not use more effectively what resources
they have, their level of service will diminish. The Rand
Corporation has provided assistance to the New York City Fire Department and
others over the past decade, concentrating on deployment policies,
which tie available resources to their distribution and movement in
the field. This lecture will first provide a brief overview of the
deployment policies that have been analyzed with the help of
mathematical models. This will be followed by a more complete discussion of
one of these policy questions: how should available fire companies
be temporarily relocated to provide coverage when many other
companies are busy fighting large fires?
I. INTRODUCTION
Fire is the leading cause of catastrophic accidents (those in which five
or more people die) in the United States, annually destroys over $4 billion in
property, and costs the total economy an estimated $14 billion per year.
Moreover, some of the underlying problems are becoming more severe. For example,
technological change has created new fire risks, deterioration of inner-city
neighborhoods has spawned rising numbers of fires, and the incidence of arson
fires has risen dramatically.
At the same time, fire departments in many cities have experienced budget
reductions or growth at a rate less than the rate of inflation. If they are
unable to use what resources they have*more effectively than in the past, their
level of service must diminish.
Improved effectiveness, however, is not easy to achieve in fire
departments. Fire service management and operations lean heavily on tradition and
on rules of thumb, many of which have not changed very much since the turn of
1980 Mathematics Subject Classification 90C50.
Copyright © 1981 American Mathematical Society
175

176
WARREN E. WALKER
the century. This is especially true of fire department deployment policies,
which tie available resources to the actual distribution and use of firefight-
ing services in the field. For example, the present number and arrangement of
fire companies in most cities are based more on historical factors, such as
where volunteer companies were first organized, than on a careful analysis of
actual needs. In 1968, a seven-year, multimillion dollar effort was begun that
was primarily devoted to developing and testing new models and methods for fire
department deployment analysis. The work was performed at The New York City-
Rand Institute. The results have been used in over fifty cities throughout the
United States [2].
The general subject of deployment analysis includes a variety of topics,
some of which can be addressed using mathematical models and others not. Most
of the issues that can be analyzed using mathematical models concern the manner
in which fire companies are located and dispatched:
o How many fire companies should be on duty? This may be a planning
decision related to the department's budget, or it may concern the
appropriate variation in company levels by time of day or by season of
the year.
o How many fire companies should be allocated to each region of the city?
A simple nonlinear programming model called the Parametric Allocation
Model was developed to address this question [4].
o Where should the city's fire companies be located? This question
refers to choosing sites for fire stations. A descriptive deterministic
model that calculates expected travel times to fires for different
arrangements of fire companies was developed. The model is called the
Firehouse Site Evaluation Model [10J.
o How many fire companies should be sent to an incoming alarm? The
answer may depend on the availability of companies at the time of the
alarm and what is known about the nature of the incident at the time
the dispatch is made. A semi-Markovian decision model was developed
that considers explicitly information on the current alarm and expected
future events [9]. The model Assumes that alarms occur and are
extinguished according to random processes. Its state space is the
number of companies busy and the potential seriousness of the incoming
alarm. The decision variable is the number of companies to dispatch,
and the objective function measures travel time to serious fires.
o Which particular fire companies should be dispatched? Most fire
departments dispatch the units that are closest to the location of an
Numbers in square brackets identify references at the end of this paper.

MATHEMATICAL MODELING APPLIED TO THE RELOCATION OF FIRE COMPANIES 177
incident. However, using insights from a queueing model, it was found
that under certain conditions it pays not to dispatch the closest
unit [1].
o Which fire companies should be temporarily relocated when a large fire
depletes one part of the city of its fire protection? When one large
fire or several small fires are being fought in a single region of a
city, protection against future fires in the same region is considerably
reduced. It is standard practice in most urban fire departments to
protect the exposed region by temporarily relocating fire companies
from outside the region into some of the vacant firehouses within the
region [6],
All of the above issues are discussed in detail in [3]. The remainder of this
talk will be devoted to a description of how we dealt .with the last issue.
II. THE RELOCATION PROBLEM
Figure 1 presents an illustration of a situation that is, unfortunately,
not unusual in New York City and other large cities. Two serious fires break
out at about the same time in the South Bronx. Seven ladder companies are
involved for several hours fighting the two blazes. This results in a large
region being left without a ladder company close by to respond if another fire
should break out in the area.
When this happens in a large city the fire department usually temporarily
relocates (moves) some fire companies from their firehouses in parts of the city
that are still adequately protected to some of the empty houses. In smaller
cities it may be necessary to achieve the same effect by borrowing companies
temporarily from neighboring communities via a mutual assistance agreement.
The purpose of such temporary relocations is clear—to spread out the still
available firefighting resources in order to reduce and balance the risks and
consequences that would result if other fires occur.
Any relocation method must provide answers to the following four
questions :
1. In what situations are relocations to be made?
2. Which of the empty firehouses should be filled?
3. Which of the available fire companies should be moved?
4. To which empty houses should each be moved?
When alarm rates are low, the fires requiring relocations of firefighting
units are rare. They occur one at a time, and when they occur, no other fires
are typically in progress. Thus, under low alarm rate conditions it is

178
WARREN E. WALKER
iO " Uncovered "
Region
O
• - Empty firehouses
O ■ Firehouses with companies in quarters
A - Companies in quarters but reserved
ML - Location of fire
Fig. 1 - A sample relocation problem

MATHEMATICAL MODELING APPLIED TO THE RELOCATION OF FIRE COMPANIES 179
possible to plan in advance for relocations with a reasonable expectation that
the plan will be able to be carried out. Experienced fire officers imagine a
hypothetical incident, say a three-alarm fire at a particular alarm box. Using
their judgment, and assuming that all other fire companies not called to the
third alarm will be available, they formulate a specific relocation plan. The
plan consists of a list of temporary transfers of engine companies and ladder
companies. Figure 2 is an example of an alarm assignment card used for
dispatching and relocating fire companies in New York City. There is one of these
cards for every alarm box in the city (over 14,000). The left side of the card
shows the units (engine companies, ladder companies, and battalion chiefs) that
are to be dispatched to alarms received from that box. The right hand side
shows the companies that are to relocate when there is a serious fire in
progress near that location. For example, the second line (corresponding to a
"two-alarm" fire) shows that Engine 50 would be moved into the house of Engine
75, Ladder 49 would be moved into the house of Ladder 33, etc. When the alarm
rate is low, alarm assignment cards work well. When the alarm rate is high,
however, the plans often break down. The reason for the breakdown is simply
that at high alarm rates several incidents (including small fires) may be in
progress simultaneously, and the officers who created a relocation plan for one
particular incident could not have anticipated this. To make a good and imple-
mentable relocation in this situation requires knowledge of the status of all
the fire companies at the department's disposal and the nature of all incidents
in progress at the time action must be taken. There are so many possible
variations of the situation that can be encountered that there is no way to do this
in advance.
Figure 3 shows two of the problems that can arise with pre-planned
relocations .
1. A company that is supposed to relocate is already busy.
2. A company that is supposed to relocate is available, but moving it
might create an even bigger gap in coverage.
We set out to design a new relocation procedure that could be used as part of
an on-line real-time computer-assisted command and control system and that
could be relied on to produce relocations that we, the fire department, and the
public would agree were "good" (as opposed to "optimal") in all types of
situations.
Our approach to developing a procedure was to view each of the questions
raised above as a separate decision problem, with the solution to each problem
being used as input to the next problem. We did this because the overall
problem is a very large multiobjective problem that would be hard to implement

3311
CRESTON AVENUE and 192nd STREET
ENGINE CO'S
48 75 79
81 88 42
L) U6 62 95
92 hb 66 93
82 71 60 69
c.
Rm.
3
[LADDER
CO'S
33 37
46
38
27
36
7 !
•. c.
19
18
Special
C«v*r-
B.C.
' 15
D. C.
6
B
R0NI
COMPANIES TO CHANCE LOCATION .
ENGINE
j 50-75 38-79
41-46 90-62
67-95 1
83-92 94-45
80-68 59-93 |
96-82 35-71
53-60 40-69
LADDE*
49-33
32-37
19-27
3^-36
Fig. 2 - Alarm assignment card for Bronx box 3311

MATHEMATICAL MODELING APPLIED TO THE RELOCATION OF FIRE COMPANIES 181
• Available
O Unavailable
5fc Location of fire
Fig. 3-*-Problems with the traditional relocation policy

182
WARREN E. WALKER
within the computer time and space constraints of a real-time system (which
would be performing other functions at the same time).
Most of the objectives to be satisfied by a "good" solution to the
relocation problem were difficult to quantify. For example, in choosing a fire
company to be moved, one would not want to move a company that was "too busy,"
that was protecting "too large" an area, or that would have to travel "too far"
from its own house. In deciding which houses should be filled and which should
remain empty one is faced with a conflict between efficiency and equity. The
department would like to place its fire companies near where the fires are
expected to occur, so they can reach them as fast as possible. However, they
must provide an acceptable level of fire protection to all regions of the city
—even those regions where the actual incidence of fires is low.
Separating the problem into four stages, each with its own objective func-
tion, made it easier for us to take all of the objectives into account.
III. STAGE 1; WHEN TO RELOCATE
This question can be translated into the question, "when is an area under-
protected?" One way of obtaining an operational answer is to set a minimum
coverage standard (e.g., maximum travel time or travel distance) for every point
in the city, based on the firefighting demands in the area. In practice,
however, it would be difficult (and rather arbitrary) to specify minimum coverage
standards. An attractive alternative is to let the way firefighting units are
already allocated to areas implicitly define the minimum coverage standards for
those areas. Usually fire companies are not uniformly distributed over a city
but are concentrated in some areas and spread out in others. This distribution
is the result of complex forces—some political, some operational, others
historical. In working with its existing distribution of resources, a fire
department has implicitly decided how it wishes to balance equity against
efficiency, at least in the short run. In the long run, of course, the fire
department may modify the distribution by building new firehouses. By assuming that
the department is satisfied with the distribution of fire companies, we can
define a minimum coverage standard for the relocation problem that will maintain
approximately the same relative geographic distribution of fire companies as
currently exists. This, in turn, will maintain approximately the same
variation in travel times (or distances) as currently exists between the areas.
Therefore, the coverage criterion that we used required that for every
alarm box in the city, at least one of the k closest engine houses and at least
one of the p closest ladder houses contain an available company, k and p are
parameters that are set by fire department policy. New York City has used
Complete details of the algorithm are contained in [6].

MATHEMATICAL MODELING APPLIED TO THE RELOCATION OF FIRE COMPANIES 183
k - p = 2. Relocations are recommended whenever this coverage criterion is
violated.
The application of this criterion is simplified by noticing that many
alarm boxes will generally have the same k closest engines or p closest ladders.
We call the aggregate of all alarm boxes having the same k closest engines an
engine response neighborhood (written "engine RN" for brevity). A ladder
response neighborhood ("ladder RN") is defined as the set of alarm boxes having
the same p closest ladders. The set of engines and ladder response
neighborhoods each form nonoverlapping partitions of the city. They are defined
separately since the coverage standard is to be applied separately to engines and
ladders in order to keep a balance of each unit type in each region. The
definition of minimum coverage can now be restated as: there must be no engine
response neighborhood with all of its k engines unavailable and no ladder response
neighborhood with all of its p ladders unavailable.
The use of response neighborhoods considerably reduces the calculations
required to check on coverage. For example, in the Bronx there are over 2000
alarm boxes but with p = 2 fewer than 50 ladder RNs. Figure 4 shows the ladder
RNs in the Bronx. Note that in regions where the ladder companies are close
together the RNs are small, and where the companies are far apart they are larger.
We have delayed until now giving a precise definition of unit
"availability" for purposes of minimum coverage. It makes no sense to relocate a unit
into the house of a company responding to (but not yet working at) an alarm,
returning from an alarm, or due back soon from a working fire. Therefore, we
consider a company to be unavailable only if it is working at a fire expected
to last for a "considerable" length of time (in practice, more than one hour).
IV. STAGE 2: WHICH HOUSES TO FILL
The primary objective of the relocation algorithm is to maintain minimum
coverage as just defined. It makes sense to do this by moving as few companies
as possible, since moving companies increases communication problems, places
them in regions with which they may not be familiar, and takes them away from
their home bases, food, and dry clothes. So we take as our criterion for the
determination of empty houses to fill: have every response neighborhood
covered, but move as few companies as possible. This translates into an integer
program known as the set covering problem.
Suppose there are K uncovered RNs and L vacant houses whose busy companies
cover or serve these RNs. For our decision variables let x. = 1 if house j is
J
to be filled and x. = 0 otherwise. Then the problem is:

184
WARREN E. WALKER
Response neighborhoods
shaded
Legend: 47-54 means ladder 47 is first-due
and ladder 54 is second-due. A
response neighborhood corresponds
to two ladders, independent of
their arrival order.
Fig. 4 — Ladder response neighborhoods in the Bronx

MATHEMATICAL MODELING APPLIED TO THE RELOCATION OF FIRE COMPANIES 185
L
minimize E x.
subject to Z a x. >_ 1
j-1 ±J J i = 1, 2, .... K
x = 0, 1 j - 1, 2, ..., L
jl if the jth house's busy company covers
where a.. = : or serves the ith RN,
/ 0 otherwise
The matrix of the a.. is known as the incidence matrix for the covering problem.
Each row of the matrix corresponds to an uncovered RN. There will be p
elements equal to 1 in each row when we consider ladders , and k equal to 1 in
each row when we consider engines. The columns correspond to the houses of the
unavailable companies. There will be a 1 in each row-column position that
marks the correspondence between an unavailable company and a RN it is supposed
to cover when it is available. The output of stage 2 is a set of M (M <_ L)
vacant houses to be filled, which is the input to stage 3.
This problem can, of course, be solved exactly. However, for reasons of
speed and computer storage requirements, we developed simple heuristic
procedures for solution of this problem, and those encountered in the next two
stages. In testing the algorithms, the results obtained using exact procedures
were compared to those obtained using the heuristics. Fortunately, in the
tests the optimal solution was always obtained using the heuristic methods.
The basic heuristic rule for selection of a house to fill is to select
first the house associated with the largest number of uncovered RNs. After
application of this rule, the covering problem is reduced by the elimination of
the house just selected to be filled and all RNs that will be covered as a
consequence of filling it. In the same way, another house is selected to be
covered. This procedure continues until all RNs are covered. The rule may be
applied several times using alternate starting points, and the best of the
results chosen.
V. STAGE 3: WHICH AVAILABLE COMPANIES TO MOVE
Once the houses to be filled have been selected on the basis of the minimum
coverage criterion (which is basically an equity criterion), there may be many
available companies that could be moved into those houses. Of course, no
company should be moved if, by moving it, the minimum coverage criterion is
violated. We therefore only consider moves that do not violate the standard.
In addition, we wish to apply the following secondary "efficiency" criteria:

186
WARREN E. WALKER
1. Do not move a company "too long" a distance.
2. Do not move a company that is "too busy."
3. Do not move a company that is protecting "too big" an area.
The way to tradeoff among these criteria is not immediately clear. A function
that measures travel time was found to take all these secondary factors into
account. We end up with a mathematical programming problem whose objective is
to minimize the total expected travel time to alarms that occur in the regions
affected by the moves. As in stage 3, the mathematical programming problem is
solved heuristically.
The following simplified scenario provides the background for the
development of the objective function. Referring to Fig. 5, suppose Ladder 31 has
just responded to a serious fire. Its house is now empty and we wish to
evaluate possible relocations into it. The houses of Ladder Companies 37 and 38 are
currently covered, and either one may be moved into Ladder 31fs house. We want
to evaluate which move is superior and if indeed any move should be made.
First, consider the data required to make the evaluation. Let R _ denote
the region in which Ladder 31 would be the closest company to all alarm boxes
if it were available in its house. We must take into account which of Ladder
31fs neighbors are currently available when we decide what constitutes R ...
Let A- and A denote respectively the physical area and the alarm rate of this
region. (Note that this region changes as the pattern of available and busy
companies changes.) Let R~_ denote the region in which Ladder 37 is currently
the closest available company, and A._ and A. be, respectively, the area and
alarm rate in R^7» Once again, in deciding what constitutes R- we must take
into account which of Ladder 37fs neighbors are available. Similar definitions
apply for R~0, A._, and A00. We assume that the alarm arrivals are a Poisson
Jo Jo Jo
process and that all the above parameters have been estimated. In addition,
r..» the time required to relocate a company from location i to location j, is
assumed to be known.
The expected travel time of the first-arriving company to an alarm in the
regions served by these companies depends on which companies are available to
respond to the alarm. If the closest company is not in quarters, the second
closest company must play the role of the closest company, and consequently the
travel time will be longer. The computational requirements for the exact
calculation of expected travel times in such a dynamically changing environment
*
are formidable. We therefore used an approximate method based on two models :
The theoretical and empirical justification for the use of these models
is provided in [5] and [7].

MATHEMATICAL MODELING APPLIED TO THE RELOCATION OF FIRE COMPANIES 187
First Due
Areas
L31
Empty House
(To Be Filled\
Other Ladder Company
Locations
Fig. 5 - A simple relocation problem: Which of the two
available companies should fill the empty house?

188
WARREN E. WALKER
1. average travel distances in a region are directly proportional to the
square-root of the area of the region and inversely proportional to the
number of companies located in the region; and
2. average travel times in a region increase linearly with the average
travel distance.
The travel-distance model estimates the regional average travel distance
for the first-arriving unit as c_ /A/N when the region has area A and has N units
available. To estimate the average second-arriving travel distance c. is
replaced by c~. In our case N = 1, so the expected travel distance of the closest
responding unit in R. when Company i is available is estimated by c,/aT. If it
is unavailable, but its neighbors are available (as is more or less the case if
minimum coverage is being guaranteed), then c9/a7 is the expected response
distance of the closest responding unit. Denoting the average response velocity in
Ri by v., we have t (i) = c 47/v as the expected travel time to alarms in R.
if Company i is available, and t (i) = c?/k7/v. as the expected travel time to
alarms in R. if Company i is busy.
If the alarms in R. are arriving according to a Poisson process with an
average of A. alarms per hour, then A.T alarms would occur on the average in
the region during a period of length T hours. Ignoring some of the complicated
dynamic behavior that could occur in R. during the time a large fire is in
progress, we have t.A.T or t A.T as the expected total first-arriving travel time to
alarms occuring in R. during the interval T—the duration of the fire that is
causing the relocation problem.
We now return to the simple scenario of Fig. 5 where Ladder Companies 37
and 38 are candidates to relocate into Ladder 31fs house, and calculate the cost
of relocating Ladder 37 in terms of expected total travel time. The relevant
information for this calculation is given in the table below.
i
37
38
31
ri.31
(hours)
0.2
0.1
A.
l
(alarms/hr.)
0.2
1.2
1.7
A.
l
(sq. miles)
0.9
1.3
0.9
Tx(i)
(mins.)
1.7
2.1
1.8
T2(i)
(mins.)
2.8
3.5
2.9
Assume that the fire at which Ladder 31 is working will last one hour (i.e.,
T = 1 hour). In order to compare the cost of relocating Ladder Company 37 to
the cost of relocating Ladder Company 38 we will consider alarms that occur in
the interval T1 = [0, T + max(r0., 01 ,r00 ..)] = [0, 1.2]. That is, Tf has to
J/,Ji Jo,jl
be long enough to encompass all of the effects of any relocations in the system.

MATHEMATICAL MODELING APPLIED TO THE RELOCATION OF FIRE COMPANIES 189
The cost of moving Ladder 37 to Ladder 31 (c _ --(T1)) is based on the
assumption that Ladder 37 spends a time r__ „. traveling to Ladder 31!s house,
stays at that house until Ladder 31 returns from the fire at time T, and then
returns home. So R is covered by a second-closest company during the interval
[0,r^7 - ] and by a closest company during the interval [r-_ ^ T1]; R~7 is
covered by a second-closest company during the interval [0,Tf]; and ROQ is
Jo
covered by a closest company during the interval [0,Tf]« We are using an
important property of the Poisson process, namely, that when two or more
independent processes are observed simultaneously, the "joint" process (the results
of both taken together) is also a Poisson process and has as its rate the sum
of the rates of the individual processes. The components of the cost are:
For Ladder 37fs area: T2(37)A3?Tf = (2.8)(0.2)(1.2) = 0.67
For Ladder 31's area: T2(31)X3ir37 31 + t^ODA^T' - r^ ^)
= (2.9)(1.7)(0.2) + (1.8)(1.7)(1.0)' = 4.05
For Ladder 38fs area: T.(38)AOQTf - (2.1)(1.2)(1.2) = 3.02
1 Jo
Total cost (minutes) = c qiC1"') = 7*7^
If Ladder 38 relocated into Ladder 31's house, the expected total first-
arriving ladder travel time over the interval [0,T'] would be calculated in a
similar fashion. The result is that c00 0-(Tf) = 9.14 minutes, which is sig-
Jo, Jl
nificantly higher than c-_ -_(Tf). The cost of making no relocation can also
be calculated. In this case the cost would be 9.35 minutes. So, in this case,
the best policy would be to relocate Ladder 37 into the house of Ladder 31.
In general, if we let a - \1'r\/v,, we can show that the cost (in expected
total travel time) of relocating available company i into the empty house of
company j is given by
c..(T) = (c2- cl} {ai(I + ry) + V-J.
and the cost of making no relocation is just (c? - c.)a.T.
Notice that each of the three secondary criteria—relocation travel
distance (r..), the ffbusyness" of a company (A.), and the size of the region
protected by a company (A.)—are all explicitly included in the cost function.
In addition, another element appears that perhaps had not been anticipated:
the duration of the fire causing the relocation problem. According to the
cost function, it is possible that a different relocation would be suggested
for a short incident than for a long incident. In fact, using this function
it is possible to determine what the predicted length of the incident must be
before it becomes advantageous to relocate.

190
WARREN E. WALKER
Figure 6 illustrates the typical relocation costs for the situation in
which Ladder Company 31 is working at a fire and Ladder Companies 37 and 38 are
available to relocate. The average total first-arriving ladder travel time is
shown for fires that occur during the duration of the incident leading to the
relocation for four alternatives:
1. No relocation (Ladder 31's house remains uncovered);
2. Move Ladder 38, which is closer to Ladder 31, but is a busy company;
3. Move Ladder 37, which is farther away from Ladder 31, but is less
busy;
4. Relocate Ladder 37 into Ladder 38's house, and relocate Ladder 38 into
Ladder 31—called a "successive moveup."
For any given value of T, the best relocation is the one for which the function
is the smallest. Examination of the graph indicates that it certainly does not
pay to make a relocation for an incident of predicted duration less than 15
minutes. If the fire will last longer than that, the best plan is to move
Ladder 37 into Ladder 31's house. If the incident lasts more than a half hour,
there is a clear advantage to this relocation. Note that if Ladder 37 could
not be moved it would not be worthwhile to make any relocation for a fire
lasting a half hour or less. Figure 6 also shows that the successive moveup of
L37 to L38 to L31 is slightly worse than just relocating Ladder 37 to Ladder
31's house.
Of course, these observations depend on the characteristics of the
particular problem we have been examining. Nevertheless, examination of cost
functions for many situations suggests some generalizations. First, "successive
moveups" have about the same travel-time cost as simple relocations. They are
sometimes a little better, more often a little worse. But they also move twice
as many companies, thereby increasing the inconvenience to the men—as well as
increasing communication and control problems. For these reasons, and after
consultation with fire department senior officers, successive moveups were
eliminated from further consideration in New York City. Second, in most
situations in the city there seemed to be a clear travel-time advantage to relocating
only if the relocation were to last more than one hour. (In the example, the
advantage showed at about a half hour, but this is due to the fact that Ladder
31 was at that time the busiest ladder company in the city.) In general, an
incident that will last an hour or more is identifiable by a chief when he
first arrives at the fire. So, rather than requiring the exact duration of all
incidents (the value of T in the cost function), it was suggested that
relocations be made only for fires expected to last more than one hour.

MATHEMATICAL MODELING APPLIED TO THE RELOCATION OF FIRE COMPANIES 191
No Relocation
V.
Ladder L 38 + L31
>
2
H
M
cd
9.0
i 8'°
Ladder L37 -* L38 -* L31
>i
Ladder L37-* L31
cd
4J
o
H
<D
GO
cd
U
<D
3
7.0J- •
0.0*—
0.5 1.0 1.5 2.0 2.5
Incident Duration (hours) T
Fig. 6—A comparison of relocations into the house of Ladder 31

192
WARREN E. WALKER
The actual duration of such "serious fires" does not change the relative
travel-time cost rankings very much. That is to say, if moving Ladder 37 to
the Ladder 31 house looks better than moving Ladder 38 there for a one-hour
fire, it also looks better for a two-hour fire. This is important since, if it
were not true, the identities of the relocating companies would depend on
accurate predictions of fire duration (which are not generally easy to make). In
the method we developed, travel-time cost calculations are based on the average
duration of a serious fire—about one hour.
The mathematical formulation of the stage three problem (which available
companies to move) is similar in structure to a "transportation" problem with
additional constraints to assure that the coverage criteria are not violated.
We let j = 1, 2, ..., M refer to the empty houses to be filled,
j=M+l, ..., M + N refer to the available companies, and k = 1, 2, ..., L
refer to the RNs associated with the available companies. The objective
function to be minimized is the total expected travel-time during the relocation
incident. If we fill empty house j with available company i.(j = 1, 2, ..., M),
the total cost of this set of moves is
M
M
E c. . = (c_ - c.) £ {a. (T + r. .) + a.r. .}
. , i.j 2 1 . . i. i.j j i.j
J=l J J-l J J J
The decisions are framed in terms of which available company is assigned
to which empty house. The decision variables will be x.., where x.. = 1 if
ij* ij
available company i is assigned to relocate into empty house j and x.. = 0
otherwise. A "dummy" empty house (j = 0) is used so that if available company
i is to remain in its own house, x.n = 1. The integer linear program to be
solved is:
M MfN
min. £ £ cJfx..
j-l i-Mfl ±J 1J
M4-N
s.t. E x.. = 1
i=M4-l 1J
j = 1. 2, ..., M
£ x.. = 1
j=0 ^
i = M4-1, M4-2, . . . , M4-N
M4-N
tt«J M
E a.. x.n + E E a., x. . > 1
i-MH lk l0 i-MU j-l Jk 1J ~
Xij = °- *
k = 1, 2, ..., L
for all i, j.

MATHEMATICAL MODELING APPLIED TO THE RELOCATION OF FIRE COMPANIES 193
The objective function and the first two sets of constraints have the
structure of a transportation problem. The first set of constraints requires
that all M of the empty houses are filled. The second set of constraints
guarantees that all available companies are assigned somewhere. As in stage 2, the
coefficients a form the incidence matrix between firehouses and response
neighborhoods. a = 1 if available company i serves response neighborhood k
and is zero otherwise. The last set of constraints requires that none of the
RNs that were being covered by available companies be uncovered.
Rather than solve such a large integer programming problem exactly, we use
a heuristic algorithm. The heuristic rule that we use for determining the
available company to move into a given house is to try to fill each empty house
with the available company having the lowest relocation cost associated with
the move. The relocation costs are the c described above. To help select
the relocatees (companies to be relocated) a ranked list of candidate relo-
catees can be created for each house to be filled. The companies on the list
are the available ones, ordered by their c values.
Each move must be checked against the coverage criterion to assure that no
RNs become uncovered. A company on any relocatee list may be relocated without
violating minimum coverage. But, if the selections are made independently for
each vacant house to be filled, the resulting set of moves might have the same
company moving into more than one house, or might leave one or more RNs
uncovered by moving neighboring companies.
A feasible relocation is generated by successive applications of the
heuristic and the feasibility test. The procedure begins with one house to be
filled and progresses in sequence through the others one by one. If the lowest
cost move for each house produces a feasible relocation, that relocation is
optimal and no further computations are necessary. Otherwise, since the
algorithm is fast, several feasible relocations are produced by changing the order
in which houses being filled are considered, and by changing the heuristic for
the first house being considered to "choose the available company with the qth
lowest relocation cost." The least-cost relocation generated after all trials
is used as the stage 3 solution.
VI. STAGE 4: SPECIFIC RELOCATION ASSIGNMENTS
The output of stage 3 is a specific set of assignments or "moves" of
available companies to the empty houses being filled. However, the assignments
sometimes make the relocating companies travel further than another possible
assignment of the same set of companies to the same empty houses. In some rare
instances the assignment can even make relocating companies travel on paths
that cross. This is because travel distance is only one of the components of

194
WARREN E. WALKER
c . While the relocating travel distance matters, it is actually distance
times alarm rate times the square-root of area that is being considered, and so
the algorithm can sacrifice relocation travel distance for gains in expected
travel times to alarms.
Yet for several reasons fire departments are concerned with the distance
that relocating companies must move. One reason is that shorter relocation
distances mean less of a burden on relocating companies and larger availability
times. Another is that keeping the relocation distance down tends to keep
companies in areas where they are familiar with street patterns as well as with
particular firefighting problems. To solve this problem we view stage 3 as a
device for selecting the companies to relocate and ignore the specific moves it
suggests. We then determine the specific assignments that minimize total
travel distance. Of course this would not make sense if the overall relocation
cost were much higher. But in most cases the resulting "reassignment"
increases the relocation cost very little, and can significantly reduce the total
distance traveled.
We now let the index j = 1, 2, ..., M refer to the M empty houses selected
by stage 2 and the index i = 1, 2, ..., M refer to the M available companies
selected by stage 3. Again, r.. denotes the time required for a unit to
relocate from "full" house i to "empty" house j, and the decision variable x.. = 1
if available company i is assigned to empty house j and is zero otherwise.
Mathematically, stage 4 involves solving a traditional assignment problem:
Find {x..} to
min.
M
V
M
I
r. . x..
s.t. £ x = 1 i=l,2,...,M
j-l
M
I x.. = 1 j = 1, 2, ..., M
i=l 1J
x.. = 0, 1.
For small problems, say up to M s 5, solutions can be obtained quickly by
complete enumeration of all M! permutations (5! = 120). In fact it would be
unusual to have to make even five relocations at one time. Usually a fire
progresses through a number of stages (first alarm, second alarm, third alarm, etc.)

MATHEMATICAL MODELING APPLIED TO THE RELOCATION OF FIRE COMPANIES 195
At each stage the number of relocations needed would be small. For example,
the hypothetical case of two serious fires that break out simultaneously in the
South Bronx, which was presented in Section II, led to the need for only four
relocations. Figure 7 shows both the least cost set of relocations resulting
from stage 3 of the algorithm (the dotted lines), and the least travel distance
solution produced by stage 4 (the solid lines). The solution produced by
stage 4 results in a reduction of 22 percent in travel distance with only a 9
percent increase in the cost function.
VI. TESTING AND IMPLEMENTATION
Before the New York City Fire Department would consider using the
relocation algorithm in its Management Information and Control System, it insisted on
extensive testing. We subjected the algorithm to a number of tests, first with
problems that were designed to present difficult or interesting situations,
second in a simulation model in which over 3600 alarms were generated at random
according to historical patterns, and third, to provide a strenuous realistic
test, in specifying relocations for one of the worst evenings ever experienced
in the Bronx. In the last test, the sequence of incidents was reconstructed and
a simulation was run to determine what would have occurred if the relocation
algorithm had been operating during one of the most trying periods in recent
departmental history. Finally, the algorithm was run in parallel with the
existing manual system in one of the fire department's communications offices.
The problem whose heuristic solution is shown in Fig. 7 was also solved
using an exact integer programming computer code. The result obtained was
identical to the minimum cost solution found by the heuristic algorithm.
However, the heuristic required only one-quarter of the CPU time and only one-half
the amount of computer core storage.
When the algorithm was tested using a simulation model that recreated the
actual situation in the Bronx on July 4, 1969, it was found that, if it had
been operating at the time, it would have avoided almost all of the problems
that actually occurred that evening. For example, in the actual situation at
one point in the evening only 79 percent of the alarm boxes in the Bronx had at
least one of their two closest ladder companies available in quarters. Using
the algorithm over 90 percent of the boxes did. Of course, the algorithm never
leaves a response neighborhood without "minimum coverage." However, on that
night, a total of 16 RNs were actually left uncovered for periods ranging from
30 minutes to 1.6 hours. In addition, the algorithm generated its relocations
gradually and continually over time, while relocations made by the dispatchers
were generally made in spurts. For example, at one point both methods had
produced 23 relocations but the algorithm had called for relocations to be made at

196
WARREN E. WALKER
o •
• -
o -
A -
A
—
Empty firehouses
Firehouses with companies in quarters
Companies in quarters but reserved
Location of fire
Recommended Solution
Least Cost Solution
Fig. 7 - A comparison of the solutions from stages 3 and 4

MATHEMATICAL MODELING APPLIED TO THE RELOCATION OF FIRE COMPANIES 197
16 separate times, while the dispatchers had made their relocations at only five
different times, (Ten of their relocations were made at one time.)
The relocation algorithm was implemented as part of the New York City Fire
Department's real-time computer-based Management Information and Control System
in the middle of 1977. In addition to recommending relocations, the MICS
recommends dispatches, maintains the status of fire companies and alarms in progress,
and updates statistical records. The system, which was first implemented in
the Brooklyn communications office, has now been expanded citywide [8].

198
WARREN E. WALKER
REFERENCES
1. Carter, Grace, Jan Chaiken, and Edward Ignall, "Response Areas for Two
Emergency Units," Operations Research, Vol. 20, No. 3, 571-594 (1972).
2. Chaiken, Jan, "Transfer of Emergency Service Deployment Models to
Operating Agencies," Management Science, Vol. 24, No. 7, 719-731 (1978).
3. The Rand Fire Project (Warren Walker, Jan Chaiken, and Edward Ignall,
editors), Fire Department Deployment Analysis, Elsevier North-Holland,
New York (1979).
4. Rider, Kenneth, "A Parametric Allocation Model for the Allocation of Fire
Companies in New York City, Management Science, Vol. 23, No. 2, 146-158
(1976).
5. Kolesar, Peter, and Edward Blum, "Square Root Laws for Fire Engine
Response Distances," Management Science, Vol. 19, No. 12, 1368-1378 (1973).
6. Kolesar, Peter, and Warren Walker, "An Algorithm for the Dynamic
Relocation of Fire Companies," Operations Research, Vol. 22, No. 2, 249-274
(1974).
7. Kolesar, Peter, Warren Walker, and Jack Hausner, "Determining the Relation
Between Fire Engine Travel Times and Travel Distances in New York City,"
Operations Research, Vol. 23, No. 4, 614-627 (1975).
8. Mohan, John J., "Starfire, F.D.N.Y.," With New York Firemen, Vol. 41, No. 1,
12-13 (1980).
9. Swersey, Arthur, "A Markovian Decision Model for Deciding How Many Units
to Dispatch," in Models for Reducing Fire Engine Response Times, Ph.D.
Thesis, Columbia University, Hew York (1972).
10. Walker, Warren, Firehouse Site Evaluation Model: Executive Summary,
Report R-1618/1-HUD, The Rand Corporation, Santa Monica (1975).
BCDEFGHIJ-AMS-89876543